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1l6nnts5x | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let S be the set of all a $$\in R$$ for which the angle between the vectors $$
\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}$$ and $$\vec{v}=\left(\log _{e} b\right) \hat{i}+2 \hat{j}+2 a\left(\log _{e} b\right) \hat{k}$$, $$(b>1)$$ is acute. Then S is equal to :</p> | [{"identifier": "A", "content": "$$\\left(-\\infty,-\\frac{4}{3}\\right)$$"}, {"identifier": "B", "content": "$$\\Phi $$"}, {"identifier": "C", "content": "$$\\left(-\\frac{4}{3}, 0\\right)$$"}, {"identifier": "D", "content": "$$\\left(\\frac{12}{7}, \\infty\\right)$$"}] | ["B"] | null | <p>$$\overrightarrow u = a({\log _e}b)\widehat i - 6\widehat j + 3\widehat k$$</p>
<p>$$\overrightarrow v = ({\log _e}b)\widehat i + 2\widehat j + 2a({\log _e}b)\widehat k$$</p>
<p>For acute angle $$\overrightarrow u \,.\,\overrightarrow v > 0$$</p>
<p>$$ \Rightarrow a{({\log _e}b)^2} - 12 + 6a({\log _e}b) > 0$$</p>... | mcq | jee-main-2022-online-28th-july-evening-shift |
1ldo522uv | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$$ be two vectors. Then which one of the following statements is TRUE ?</p> | [{"identifier": "A", "content": "Projection of $$\\vec{a}$$ on $$\\vec{b}$$ is $$\\frac{-13}{\\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction \nof $$\\vec{b}$$."}, {"identifier": "B", "content": "Projection of $$\\vec{a}$$ on $$\\vec{b}$$ is $$\\frac{13}{\\sqrt{35}}$$ and the direct... | ["A"] | null | $\begin{aligned} & \text { Projection of }\vec{a} \text { on } \vec{b} =\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \\\\ & = \frac{(5 \hat{i}-\hat{j}-3 \hat{k}) \cdot(\hat{i}+3 \hat{j}+5 \hat{k})}{\sqrt{1^2+3^2+5^2}}=\frac{5-3-15}{\sqrt{35}} \\\\ & = \frac{-13}{\sqrt{35}}\end{aligned}$
<br/><br/>Negative sign indicates tha... | mcq | jee-main-2023-online-1st-february-evening-shift |
1ldo6n5a3 | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}$$ and $$\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}$$ be three given vectors. If $$\overrightarrow{\mathrm{r}}$$ is a vector such that $$\vec{r} \times \vec{a}=\vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b}=0$$, then $$|\vec{r}|$$ is equal to :</p> | [{"identifier": "A", "content": "$$\\frac{11}{7}$$"}, {"identifier": "B", "content": "$$\\frac{11}{5} \\sqrt{2}$$"}, {"identifier": "C", "content": "$$\\frac{\\sqrt{914}}{7}$$"}, {"identifier": "D", "content": "$$\\frac{11}{7} \\sqrt{2}$$"}] | ["D"] | null | $\begin{aligned} & \vec{r} \times \vec{a}=\vec{c} \times \vec{a} \\\\ & \Rightarrow(\vec{r}-\vec{c}) \times \vec{a}=0 \Rightarrow \vec{r}-\vec{c}=\lambda \vec{a}((\vec{r}-\vec{c} ) \text{and} \overrightarrow{a} \text { are parallel }) \\\\ & \Rightarrow \vec{r}=\vec{c}+\lambda \vec{a} \\\\ & \Rightarrow \vec{r} \cdot ... | mcq | jee-main-2023-online-1st-february-evening-shift |
1ldpsrr3l | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$$, and $$\vec{b}$$ and $$\vec{c}$$ be two nonzero vectors such that $$|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$$ and $$\vec{b} \cdot \vec{c}=0$$. Consider the following two statements:</p>
<p>(A) $$|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|$$ for all $$\lambda \in \mat... | [{"identifier": "A", "content": "only (B) is correct"}, {"identifier": "B", "content": "both (A) and (B) are correct"}, {"identifier": "C", "content": "only (A) is correct"}, {"identifier": "D", "content": "neither (A) nor (B) is correct"}] | ["C"] | null | $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$
<br/><br/>$$ \Rightarrow $$ $|\vec{a}+\vec{b}+\vec{c}|^{2}=|\vec{a}+\vec{b}-\vec{c}|^{2}$
<br/><br/>$$
\begin{aligned}
& \Rightarrow |\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \vec{a}) \\\\
& =|\vec{a}|^{2}+|\vec{... | mcq | jee-main-2023-online-31st-january-morning-shift |
1ldswfhf7 | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>If the vectors $$\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k$$, $$\overrightarrow b = - 2\widehat i + 4\widehat j - 2\widehat k$$ and $$\overrightarrow c = 2\widehat i + 3\widehat j + \widehat k$$ are coplanar and the projection of $$\overrightarrow a $$ on the vector $$\overrightarrow ... | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "18"}, {"identifier": "D", "content": "6"}] | ["A"] | null | $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}-2 \hat{k}, \vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}$
<br/><br/>
Now, $\vec{a} \cdot \vec{b}=\sqrt{54} \Rightarrow \frac{-2 \lambda+4 \mu-8}{\sqrt{24}}=\sqrt{54}$
<br/><br/>
$\Rightarrow-2 \lambda+4 \mu-8=36$
<br/><br/>
$\Rightarrow 2 \mu-\lambda=22... | mcq | jee-main-2023-online-29th-january-morning-shift |
1ldv14lsr | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>The vector $$\overrightarrow a = - \widehat i + 2\widehat j + \widehat k$$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $$\overrightarrow b $$. Then the projection of $$3\overrightarrow a + \sqrt 2 \overrightarrow b $$ on $$\overrightarrow c = 5\widehat i + ... | [{"identifier": "A", "content": "$$\\sqrt6$$"}, {"identifier": "B", "content": "2$$\\sqrt3$$"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "3$$\\sqrt2$$"}] | ["D"] | null | <p>First, we write $\overrightarrow{b}$ as a linear combination of $\overrightarrow{a}$ and $\overrightarrow{j}$ since $\overrightarrow{b}$ is a rotation of $\overrightarrow{a}$ about the y-axis.</p>
<p>$\vec{b}=\lambda \vec{a}+\mu \hat{j}=\lambda(-\hat{i}+2 \hat{j}+\hat{k})+\mu \hat{j}=-\lambda \hat{i}+(2 \lambda+\mu ... | mcq | jee-main-2023-online-25th-january-morning-shift |
1ldwwcfsd | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k$$ and $$\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k$$. Let $${\overrightarrow \beta _1}$$ be parallel to $$\overrightarrow \alpha $$ and $${\overrightarrow \beta _2}$$ be perpendicular to $$\overrightarrow \alpha $$. If... | [{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "11"}] | ["B"] | null | Let $\vec{\beta}_1=\lambda \vec{\alpha}$<br/><br/>
Now $\vec{\beta}_2=\vec{\beta}-\vec{\beta}_1$<br/><br/>
$$
\begin{aligned}
& =(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})-\lambda(4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}) \\\\
& =(1-4 \lambda) \hat{\mathrm{i}}+(2-3 \lambda) \hat{\mathrm{j}... | mcq | jee-main-2023-online-24th-january-evening-shift |
lsblj5uf | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha \hat{i}+2 \alpha \hat{j}-2 \hat{k}$ is acute, is ___________. | [] | null | 5 | <p>$$\begin{aligned}
& \cos \theta=\frac{(\alpha \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \cdot(\alpha \hat{\mathrm{i}}+2 \alpha \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{\alpha^2+4+4} \sqrt{\alpha^2+4 \alpha^2+4}} \\
& \cos \theta=\frac{\alpha^2-4 \alpha-4}{\sqrt{\alpha^2+8} \sqrt{5 \alpha^2+4}} \\
& ... | integer | jee-main-2024-online-27th-january-morning-shift |
jaoe38c1lsfkf8y5 | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let a unit vector $$\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}$$ make angles $$\frac{\pi}{2}, \frac{\pi}{3}$$ and $$\frac{2 \pi}{3}$$ with the vectors $$\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}, \frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\sqrt{2}} \hat{k}$$ and $$\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j... | [{"identifier": "A", "content": "$$\\frac{11}{2}$$\n"}, {"identifier": "B", "content": "$$\\frac{5}{2}$$"}, {"identifier": "C", "content": "7"}, {"identifier": "D", "content": "9"}] | ["B"] | null | <p>Unit vector $$\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$$</p>
<p>$$\begin{aligned}
& \overrightarrow{\mathrm{p}}_1=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \overrightarrow{\mathrm{p}}_2=\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\fra... | mcq | jee-main-2024-online-29th-january-evening-shift |
lv0vxcgi | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let a unit vector which makes an angle of $$60^{\circ}$$ with $$2 \hat{i}+2 \hat{j}-\hat{k}$$ and an angle of $$45^{\circ}$$ with $$\hat{i}-\hat{k}$$ be $$\vec{C}$$. Then $$\vec{C}+\left(-\frac{1}{2} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\frac{\sqrt{2}}{3} \hat{k}\right)$$ is:</p> | [{"identifier": "A", "content": "$$-\\frac{\\sqrt{2}}{3} \\hat{i}+\\frac{\\sqrt{2}}{3} \\hat{j}+\\left(\\frac{1}{2}+\\frac{2 \\sqrt{2}}{3}\\right) \\hat{k}$$\n"}, {"identifier": "B", "content": "$$\\left(\\frac{1}{\\sqrt{3}}+\\frac{1}{2}\\right) \\hat{i}+\\left(\\frac{1}{\\sqrt{3}}-\\frac{1}{3 \\sqrt{2}}\\right) \\hat{... | ["C"] | null | <p>$$\begin{aligned}
& \text { Let } \vec{C}=a \hat{i}+b \hat{j}+c \hat{k} \\
& (a \hat{i}+b \hat{j}+c \hat{k}) \cdot(2 \hat{i}+2 \hat{j}-\hat{k})=1 \times 3 \times \frac{1}{2} \\
& 2 a+2 b-c=\frac{3}{2} \qquad \text{... (1)}\\
& (a \hat{i}+b \hat{j}+c \hat{k}) \cdot(\hat{i}-\hat{k})=1 \times \sqrt{2} \times \frac{1}{\... | mcq | jee-main-2024-online-4th-april-morning-shift |
lv2erzmn | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>For $$\lambda>0$$, let $$\theta$$ be the angle between the vectors $$\vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k}$$ and $$\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$$. If the vectors $$\vec{a}+\vec{b}$$ and $$\vec{a}-\vec{b}$$ are mutually perpendicular, then the value of (14 cos $$\theta)^2$$ is equal to</p> | [{"identifier": "A", "content": "25"}, {"identifier": "B", "content": "50"}, {"identifier": "C", "content": "20"}, {"identifier": "D", "content": "40"}] | ["A"] | null | <p>$$\begin{aligned}
& \text { Given } \vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k} \\
& \vec{b}=3 \hat{i}-\hat{j}+2 \hat{k} \\
& \vec{a}+\vec{b}=4 \hat{i}+(\lambda-1) \hat{j}-\hat{k} \\
& \vec{a}-\vec{b}=-2 \hat{i}+(\lambda+1) \hat{j}-5 \hat{k} \\
& (\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=0 \\
& -8+\lambda^2-1+5=0
\end{... | mcq | jee-main-2024-online-4th-april-evening-shift |
lv3veez6 | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+3 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}+3 \hat{j}-5 \hat{k}$$ and $$\overrightarrow{\mathrm{c}}=3 \hat{i}-\hat{j}+\lambda \hat{k}$$ be three vectors. Let $$\overrightarrow{\mathrm{r}}$$ be a unit vector along $$\vec{b}+\vec{c}$$. If $$\vec{r} \cdot \vec{a}... | [{"identifier": "A", "content": "21"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "27"}, {"identifier": "D", "content": "30"}] | ["B"] | null | <p>$$\begin{aligned}
& \vec{a}=\hat{i}+2 \hat{j}+3 \hat{k} \\
& \vec{b}=2 \hat{i}+3 \hat{j}-5 \hat{k} \\
& \vec{c}=3 \hat{i}-\hat{j}+\lambda \hat{k} \\
& \vec{b}+\vec{c}=5 \hat{i}+2 \hat{j}+(\lambda-5) \hat{k}
\end{aligned}$$</p>
<p>$$\vec{r}$$ is a unit vector along $$\vec{b}+\vec{c}$$</p>
<p>$$\therefore \quad \vec{r... | mcq | jee-main-2024-online-8th-april-evening-shift |
lv5grw8k | maths | vector-algebra | scalar-or-dot-product-of-two-vectors-and-its-applications | <p>The set of all $$\alpha$$, for which the vectors $$\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$$ and $$\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$$ are inclined at an obtuse angle for all $$t \in \mathbb{R}$$, is</p> | [{"identifier": "A", "content": "$$[0,1)$$\n"}, {"identifier": "B", "content": "$$\\left(-\\frac{4}{3}, 0\\right]$$\n"}, {"identifier": "C", "content": "$$(-2,0]$$\n"}, {"identifier": "D", "content": "$$\\left(-\\frac{4}{3}, 1\\right)$$"}] | ["B"] | null | <p>Given $$\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$$</p>
<p>and $$\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$$</p>
<p>angle between $$\vec{a}$$ and $$\vec{b}$$ is given by</p>
<p>$$\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$$</p>
<p>We have, $$\cos \theta < 0(\because$$ angle between $$\vec{a... | mcq | jee-main-2024-online-8th-april-morning-shift |
TRVqTQF2Low17LyL | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$, then : | [{"identifier": "A", "content": "$\\overrightarrow{\\mathbf{a}} \\cdot \\overrightarrow{\\mathbf{b}}=\\overrightarrow{\\mathbf{b}} \\cdot \\overrightarrow{\\mathbf{c}}=\\overrightarrow{\\mathbf{c}} \\cdot \\overrightarrow{\\mathbf{b}}=0$"}, {"identifier": "B", "content": "$\\overrightarrow{\\mathbf{a}} \\times \\overri... | ["B"] | null | If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are the sides of $\mathbf{a}$ triangle, then <br/><br/>$\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}$
<br/><br/>Since,
<br/><br/>$$
\begin{aligned}
\vec{a}+\... | mcq | aieee-2002 |
EmeP0gA9DJPiHDRu | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If $$\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$$ and the angle between $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$\pi /6$$ then $${\left( {\overrightarrow a \times \overrightarrow b } \right)^2}$$ is equal to : | [{"identifier": "A", "content": "$$48$$ "}, {"identifier": "B", "content": "$$16$$"}, {"identifier": "C", "content": "$$\\overrightarrow a $$ "}, {"identifier": "D", "content": "none of these "}] | ["B"] | null | $${\left( {\overrightarrow a \times \overrightarrow b } \right)^2} = {\left| {\overrightarrow a } \right|^2}{\left| {\overrightarrow b } \right|^2}\,\,{\sin ^2}{\pi \over 6}$$
<br><br>$$ = 16 \times 4 \times {1 \over 4} = 16$$ | mcq | aieee-2002 |
SUoKDdZSEYkrRjwJ | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If the vectors $$\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$$ and $$\widehat b = \widehat j$$ are such that $$\overrightarrow a ,\overrightarrow c $$ and $$\overrightarrow b $$ form a right handed system then $${\overrightarrow c }$$ is : | [{"identifier": "A", "content": "$$z\\widehat i - x\\widehat k$$ "}, {"identifier": "B", "content": "$$\\overrightarrow 0 $$ "}, {"identifier": "C", "content": "$$y\\widehat j$$ "}, {"identifier": "D", "content": "$$ - z\\widehat i + x\\widehat k$$ "}] | ["A"] | null | Since $$\overrightarrow a ,\overrightarrow c ,\overrightarrow b $$ form a right handed system,
<br><br>$$\therefore$$ $$\overrightarrow c = \overrightarrow b \times \overrightarrow a = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
0 & 1 & 0 \cr
x & y & z \cr
... | mcq | aieee-2002 |
cgkZdqQPALB8vGYw | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | $$\overrightarrow a = 3\widehat i - 5\widehat j$$ and $$\overrightarrow b = 6\widehat i + 3\widehat j$$ are two vectors and $$\overrightarrow c $$ is a vector such that $$\overrightarrow c = \overrightarrow a \times \overrightarrow b $$ then $$\left| {\overrightarrow a } \right|:\left| {\overrightarrow b } \right|:... | [{"identifier": "A", "content": "$$\\sqrt {34} :\\sqrt {45} :\\sqrt {39} $$ "}, {"identifier": "B", "content": "$$\\sqrt {34} :\\sqrt {45} :39$$ "}, {"identifier": "C", "content": "$$34:39:45$$ "}, {"identifier": "D", "content": "$$\\,39:35:34$$ "}] | ["B"] | null | <p>To solve this problem, let's take it step by step, beginning with calculating each of the vector magnitudes (or norms) and then finding the magnitude of the cross product vector $\overrightarrow{c}$.</p>
<p>Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are:</p>
<p>$\overrightarrow{a} = 3\widehat{i} -... | mcq | aieee-2002 |
FRP6dd3QwqsKilpn | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If $$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$ then $$\overrightarrow a + \overrightarrow b + \overrightarrow c = $$ | [{"identifier": "A", "content": "$$abc$$ "}, {"identifier": "B", "content": "$$-1$$"}, {"identifier": "C", "content": "$$0$$"}, {"identifier": "D", "content": "$$2$$"}] | ["C"] | null | Let $$\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow r .$$ Then
<br><br>$$\overrightarrow a \times \left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) = \overrightarrow a \times \overrightarrow r $$
<br><br>$$ \Rightarrow 0 + \overrightarrow a \times \overrig... | mcq | aieee-2003 |
WpRf7gPaDyKnEwUr | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | A tetrahedron has vertices at $$O(0,0,0), A(1,2,1) B(2,1,3)$$ and $$C(-1,1,2).$$ Then the angle between the faces $$OAB$$ and $$ABC$$ will be : | [{"identifier": "A", "content": "$${90^ \\circ }$$ "}, {"identifier": "B", "content": "$${\\cos ^{ - 1}}\\left( {{{19} \\over {35}}} \\right)$$ "}, {"identifier": "C", "content": "$${\\cos ^{ - 1}}\\left( {{{17} \\over {31}}} \\right)$$"}, {"identifier": "D", "content": "$${30^ \\circ }$$"}] | ["B"] | null | Vector perpendicular to the face $$OAB$$
<br><br>$$ = \overrightarrow {OA} \times \overrightarrow {OB} = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
1 & 2 & 1 \cr
2 & 1 & 3 \cr
} } \right| = 5\widehat i - \widehat j - 3\widehat k$$
<br><br>Vector perpendicu... | mcq | aieee-2003 |
mKOt6iTT5rj0k5Bl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$ and $$\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.$$ If $$\widehat n$$ is a unit vector such that $$\overrightarrow u .\widehat n = 0$$ and $$\overrightarrow v .\widehat n = 0\,\,,$$ then $$\left| {\o... | [{"identifier": "A", "content": "$$3$$ "}, {"identifier": "B", "content": "$$0$$ "}, {"identifier": "C", "content": "$$1$$ "}, {"identifier": "D", "content": "$$2$$"}] | ["A"] | null | Since, $\hat{\mathbf{n}} \perp \overrightarrow{\mathbf{u}}$ and $\hat{\mathbf{n}} \perp \overrightarrow{\mathbf{v}}$
<br/><br/>$$
\begin{aligned}
\Rightarrow \hat{\mathbf{n}} & =\frac{\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}}{|\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}}|} \\\\
\... | mcq | aieee-2003 |
8fOb3BJc4CXuZI26 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be non-zero vectors such that $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\overrightarrow a \,\,.$$ If $$\theta $$ is ... | [{"identifier": "A", "content": "$${{2\\sqrt 2 } \\over 3}$$ "}, {"identifier": "B", "content": "$${{\\sqrt 2 } \\over 3}$$"}, {"identifier": "C", "content": "$${2 \\over 3}$$ "}, {"identifier": "D", "content": "$${1 \\over 3}$$"}] | ["A"] | null | Given $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\overrightarrow a $$
<br><br>Clearly $$\overrightarrow a $$ and $$\overrightarrow b $$ are noncollinear
<br><br>$$ \Rightarrow \left( {\overr... | mcq | aieee-2004 |
bnFroekS8xy90KW5 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | For any vector $${\overrightarrow a }$$ , the value of $${\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \times \widehat j} \right)^2} + {\left( {\overrightarrow a \times \widehat k} \right)^2}$$ is equal to : | [{"identifier": "A", "content": "$$3{\\overrightarrow a ^2}$$ "}, {"identifier": "B", "content": "$${\\overrightarrow a ^2}$$"}, {"identifier": "C", "content": "$$2{\\overrightarrow a ^2}$$"}, {"identifier": "D", "content": "$$4{\\overrightarrow a ^2}$$"}] | ["C"] | null | Let $$\overrightarrow a = x\overrightarrow i + y\overrightarrow j + z\overrightarrow k $$
<br><br>$$\overrightarrow a \times \overrightarrow i = z\overrightarrow j - y\overrightarrow k $$
<br><br>$$ \Rightarrow {\left( {\overrightarrow a \times \overrightarrow i } \right)^2} = {y^2} + {z^2}$$
<br><br>Similarly, ... | mcq | aieee-2005 |
KRG2QzjpRTUqQrKh | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If $$\widehat u$$ and $$\widehat v$$ are unit vectors and $$\theta $$ is the acute angle between them, then $$2\widehat u \times 3\widehat v$$ is a unit vector for : | [{"identifier": "A", "content": "no value of $$\\theta $$ "}, {"identifier": "B", "content": "exactly one value of $$\\theta $$ "}, {"identifier": "C", "content": "exactly two values of $$\\theta $$ "}, {"identifier": "D", "content": "more than two values of $$\\theta $$ "}] | ["B"] | null | Given $$\left| {2\widehat u \times 3\widehat v} \right| = 1$$
<br><br>and $$\theta $$ is acute angle between $$\widehat u$$
<br><br>and $$\widehat v,\,\,\left| {\widehat u} \right| = 1,\,\,\left| {\widehat v} \right| = 1\,\,\,$$
<br><br>$$ \Rightarrow \,\,\,6\left| {\widehat u} \right|\left| {\widehat v} \right|\left... | mcq | aieee-2007 |
GzoMaNVU4DTaCF86 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | The vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ are not perpendicular and $$\overrightarrow c $$ and $$\overrightarrow d $$ are two vectors satisfying $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$ and $$\overrightarrow a .\overrightarrow d = 0\,\,.$$ Then... | [{"identifier": "A", "content": "$$\\overrightarrow c + \\left( {{{\\overrightarrow a .\\overrightarrow c } \\over {\\overrightarrow a .\\overrightarrow b }}} \\right)\\overrightarrow b $$ "}, {"identifier": "B", "content": "$$\\overrightarrow b + \\left( {{{\\overrightarrow b .\\overrightarrow c } \\over {\\overrigh... | ["C"] | null | $$\overrightarrow a .\overrightarrow b \ne 0,\overrightarrow a .\overrightarrow d = 0$$
<br><br>Now, $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$
<br><br>$$ \Rightarrow \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = \overrig... | mcq | aieee-2011 |
c5QQOLp2BK9AMeWV | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$.
<br/><br>Let $$\overrightarrow c $$ be a vector such that $$\left| {\overrightarrow c - \overrightarrow a } \right| = 3$$,
<br/><br>$$\left| {\left( {\overrightarrow a \times \overrightarrow b } \... | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "$${1 \\over 8}$$"}, {"identifier": "D", "content": "$${{25} \\over 8}$$"}] | ["A"] | null | Given:
<br><br>$$\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k,\,\,\overrightarrow b = \widehat i + \widehat j$$
<br><br>$$ \Rightarrow $$ $$\left| {\overrightarrow a } \right| = 3$$
<br><br>$$ \therefore $$ $$\overrightarrow a \times \overrightarrow b = 2\widehat i - 2\widehat j ... | mcq | jee-main-2017-offline |
Pf8T6ywBp4CA8ZLZEdLkh | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | The area (in sq. units) of the parallelogram whose diagonals are along the vectors $$8\widehat i - 6\widehat j$$ and $$3\widehat i + 4\widehat j - 12\widehat k,$$ is : | [{"identifier": "A", "content": "26"}, {"identifier": "B", "content": "65"}, {"identifier": "C", "content": "20"}, {"identifier": "D", "content": "52"}] | ["B"] | null | When diagonal $${\overrightarrow {{d_1}} }$$ and $${\overrightarrow {{d_2}} }$$ are given of a parallelogram then the area of parallelogram = $${1 \over 2}\left| {\overrightarrow {{d_1}} \times \overrightarrow {{d_2}} } \right|$$
<br><br>Given, $${\overrightarrow {{d_1}} }$$ = 8$$\widehat i$$ $$-$$ 6$$\widehat j$$ + ... | mcq | jee-main-2017-online-8th-april-morning-slot |
vNlAtSKfwwTlji9dN29Ja | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If the vector $$\overrightarrow b = 3\widehat j + 4\widehat k$$ is written as the
sum of a vector $$\overrightarrow {{b_1}} ,$$ paralel to $$\overrightarrow a = \widehat i + \widehat j$$ and a vector $$\overrightarrow {{b_2}} ,$$ perpendicular to $$\overrightarrow a ,$$ then $$\overrightarrow {{b_1}} \times \over... | [{"identifier": "A", "content": "$$ - 3\\widehat i + 3\\widehat j - 9\\widehat k$$"}, {"identifier": "B", "content": "$$6\\widehat i - 6\\widehat j + {9 \\over 2}\\widehat k$$ "}, {"identifier": "C", "content": "$$ - 6\\widehat i + 6\\widehat j - {9 \\over 2}\\widehat k$$"}, {"identifier": "D", "content": "$$3\\widehat... | ["B"] | null | $$\overrightarrow {{b_1}} = {{\left( {\overrightarrow {{b_1}} .\overrightarrow a } \right)\widehat a} \over 1}$$
<br><br>= $$\left\{ {{{\left( {3\widehat j + 4\widehat k} \right).\left( {\widehat i + \widehat j} \right)} \over {\sqrt 2 }}} \right\}\left( {{{\widehat i + \widehat j} \over {\sqrt 2 }}} ... | mcq | jee-main-2017-online-9th-april-morning-slot |
MeeggVCHIAFtgLpDiij1E | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If $$\overrightarrow a ,\,\,\overrightarrow b ,$$ and $$\overrightarrow C $$ are unit vectors such that $$\overrightarrow a + 2\overrightarrow b + 2\overrightarrow c = \overrightarrow 0 ,$$ then $$\left| {\overrightarrow a \times \overrightarrow c } \right|$$ is equal to : | [{"identifier": "A", "content": "$${{\\sqrt {15} } \\over 4}$$ "}, {"identifier": "B", "content": "$${{1} \\over {4}}$$"}, {"identifier": "C", "content": "$${{15} \\over {16}}$$"}, {"identifier": "D", "content": "$${{\\sqrt {15} } \\over 16}$$"}] | ["A"] | null | Given, <br><br>
$$\overrightarrow a + 2\overrightarrow b + 2\overrightarrow c = \overrightarrow 0 $$<br><br/>
$$ \Rightarrow $$ $$\overrightarrow a + 2\overrightarrow c = - 2\overrightarrow b $$<br>
<br>
Squaring both sides,<br><br/>
$${\left| {\overrightarrow a } \right|^2} + 4\overrightarrow a .\overrightarrow... | mcq | jee-main-2018-online-15th-april-morning-slot |
t3jFwvsaHi92dFMbV9vrP | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow c = \widehat j - \widehat k$$ and a vector $$\overrightarrow b $$ be such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$ and $$\overrightarrow a .\overrightarrow b = 3.$$ Then $$\left| {\overrightarrow b } \rig... | [{"identifier": "A", "content": "$${{11} \\over 3}$$"}, {"identifier": "B", "content": "$${{11} \\over {\\sqrt 3 }}$$"}, {"identifier": "C", "content": "$$\\sqrt {{{11} \\over 3}} $$"}, {"identifier": "D", "content": "$${{\\sqrt {11} } \\over 3}$$"}] | ["C"] | null | $$ \because $$ $$\overrightarrow a $$ $$=$$ $$\widehat i + \widehat j + \widehat k \Rightarrow \left| {\overrightarrow a } \right| = \sqrt 3 $$
<br><br>& $$\overrightarrow c = \widehat j - \widehat k \Rightarrow \left| {\overrightarrow c } \right|\sqrt 2 $$
<br><br>Now, $$\overrightarrow a $$ $$ \... | mcq | jee-main-2018-online-16th-april-morning-slot |
AsjpuvwmmzcqAZgjE5Xn1 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\mathop a\limits^ \to = 3\mathop i\limits^ \wedge + 2\mathop j\limits^ \wedge + x\mathop k\limits^ \wedge $$ and $$\mathop b\limits^ \to = \mathop i\limits^ \wedge - \mathop j\limits^ \wedge + \mathop k\limits^ \wedge $$
, for some real x. Then $$\left| {\mathop a\limits^ \to \times \mathop b\limi... | [{"identifier": "A", "content": "0 < r < $$\\sqrt {{3 \\over 2}} $$"}, {"identifier": "B", "content": "$$3\\sqrt {{3 \\over 2}} < r < 5\\sqrt {{3 \\over 2}} $$"}, {"identifier": "C", "content": "$$ r \\ge 5\\sqrt {{3 \\over 2}} $$"}, {"identifier": "D", "content": "$$\\sqrt {{3 \\over 2}} < r \\le 3\\... | ["C"] | null | $$\overrightarrow a \times \overrightarrow b = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
3 & 2 & x \cr
1 & { - 1} & 1 \cr
} } \right|$$
<br><br>= (2 + x)$${\widehat i}$$ + (3 - x)$${\widehat j}$$ - 5$${\widehat k}$$
<br><br>$$\left| {\overrightarrow a \t... | mcq | jee-main-2019-online-8th-april-evening-slot |
kz4wkHygFPPLV412753rsa0w2w9jx65kz0q | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = 3\widehat i + 2\widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - 2\widehat k$$ be two vectors. If a vector perpendicular to both the vectors
$$\overrightarrow a + \overrightarrow b $$ and $$\overrightarrow a - \overrightarrow b $$ has the magnitude 12 then one... | [{"identifier": "A", "content": "$$4\\left( {2\\widehat i - 2\\widehat j - \\widehat k} \\right)$$"}, {"identifier": "B", "content": "$$4\\left( { - 2\\widehat i - 2\\widehat j + \\widehat k} \\right)$$"}, {"identifier": "C", "content": "$$4\\left( {2\\widehat i + 2\\widehat j + \\widehat k} \\right)$$"}, {"identifier"... | ["A"] | null | Required vector is $\overrightarrow r$ = $$\lambda \left( {\left( {\overline a + \overline b } \right) \times \left( {\overline a - \overline b } \right)} \right)$$<br><br>
$$ \Rightarrow \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
4 & 4 & 0 \cr
2 & 0 & 4 \cr ... | mcq | jee-main-2019-online-12th-april-morning-slot |
bOQOFNGP3tT7x6szeD3rsa0w2w9jx2gbzy3 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | The distance of the point having position vector $$ - \widehat i + 2\widehat j + 6\widehat k$$
from the straight line passing through the point
(2, 3, – 4) and parallel to the vector, $$6\widehat i + 3\widehat j - 4\widehat k$$ is : | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "$$2\\sqrt {13} $$"}, {"identifier": "D", "content": "$$4\\sqrt 3 $$"}] | ["B"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264693/exam_images/n0zytcbrpcd3kgpqtlsb.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263784/exam_images/axuiomv56i9jmm3qd8t0.webp"><source media="(max-wid... | mcq | jee-main-2019-online-10th-april-evening-slot |
3h3PJW4wWUVQxcoR3jRCO | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow \alpha = 3\widehat i + \widehat j$$ and $$\overrightarrow \beta = 2\widehat i - \widehat j + 3 \widehat k$$
. If $$\overrightarrow \beta = {\overrightarrow \beta _1} - \overrightarrow {{\beta _2}} $$,
where $${\overrightarrow \beta _1}$$
is parallel to $$\overrightarrow \alpha $$ and $$\o... | [{"identifier": "A", "content": "$$ 3\\widehat i - 9\\widehat j - 5\\widehat k$$"}, {"identifier": "B", "content": "$${1 \\over 2}$$($$ - 3\\widehat i + 9\\widehat j + 5\\widehat k$$)"}, {"identifier": "C", "content": "$$ - 3\\widehat i + 9\\widehat j + 5\\widehat k$$"}, {"identifier": "D", "content": "$${1 \\over 2}$$... | ["B"] | null | Given $$\overrightarrow \alpha = 3\widehat i + \widehat j$$<br><br>$$\overrightarrow \beta = 2\widehat i - \widehat j + 3 \widehat k$$
<br><br>$${\overrightarrow \beta _1}$$
is parallel to $$\overrightarrow \alpha $$
<br><br>$$ \therefore $$ $${\overrightarrow \beta _1}$$ = $$\lambda $$ $$\overrightarrow \alpha$... | mcq | jee-main-2019-online-9th-april-morning-slot |
Cc04Cv6xi4G4IkiaNfkHp | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,$$ $$\overrightarrow b = \widehat i + \lambda \widehat j + 4\widehat k$$ and $$\overrightarrow c = 2\widehat i + 4\widehat j + \left( {{\lambda ^2} - 1} \right)\widehat k$$ be coplanar vectors. Then the non-zero vector $$\overrightarrow a \times \ov... | [{"identifier": "A", "content": "$$ - 10\\widehat i - 5\\widehat j$$"}, {"identifier": "B", "content": "$$ - 10\\widehat i + 5\\widehat j$$"}, {"identifier": "C", "content": "$$ - 14\\widehat i + 5\\widehat j$$"}, {"identifier": "D", "content": "$$ - 14\\widehat i - 5\\widehat j$$"}] | ["B"] | null | $$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 0$$
<br><br>$$ \Rightarrow \left| {\matrix{
1 & 2 & 4 \cr
1 & \lambda & 4 \cr
2 & 4 & {{\lambda ^2} - 1} \cr
} } \right| = 0$$
<br><br>$$ \Rightarrow {\lambda ^3} - 2{\lambda ^2} - 9\lambda + 1... | mcq | jee-main-2019-online-11th-january-morning-slot |
HFYRDICglIKazRYErj7k9k2k5fl9rbl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a $$
, $$\overrightarrow b $$
and $$\overrightarrow c $$
be three unit vectors such that
<br/>$$\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0 $$. If $$\lambda = \overrightarrow a .\vec b + \vec b.\overrightarrow c + \overrightarrow c .\overrightarrow a $$ and
<br/>$$\ove... | [{"identifier": "A", "content": "$$\\left( {{3 \\over 2},3\\overrightarrow a \\times \\overrightarrow c } \\right)$$"}, {"identifier": "B", "content": "$$\\left( { - {3 \\over 2},3\\overrightarrow c \\times \\overrightarrow b } \\right)$$"}, {"identifier": "C", "content": "$$\\left( { - {3 \\over 2},3\\overrightarrow... | ["C"] | null | $$\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0 $$
<br><br>$$ \Rightarrow $$ $${\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2}$$ = 0
<br><br>$$ \Rightarrow $$ $${{{\left| {\overrightarrow a } \right|}^2}}$$ + $${{{\left| {\overrightarrow b } \right|}^2}}$$ + $${{... | mcq | jee-main-2020-online-7th-january-evening-slot |
CB4aJTXbdSnUgLtX7X7k9k2k5hiep0f | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i - 2\widehat j + \widehat k$$ and $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$ be two
vectors. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times \overrightarrow a $$ and $$\overrightarrow c .\o... | [{"identifier": "A", "content": "$$ - {1 \\over 2}$$"}, {"identifier": "B", "content": "$$ - {3 \\over 2}$$"}, {"identifier": "C", "content": "$${1 \\over 2}$$"}, {"identifier": "D", "content": "-1"}] | ["A"] | null | $$\overrightarrow a = \widehat i - 2\widehat j + \widehat k$$
<br><br>$$\overrightarrow b = \widehat i - \widehat j + \widehat k$$
<br><br>$$\left| {\overrightarrow a } \right|$$ = $$\sqrt 6 $$, $$\left| {\overrightarrow b } \right|$$ = $$\sqrt 3 $$
<br><br>and $${\overrightarrow a .\overrightarrow b }$$ = 4
<br><br>... | mcq | jee-main-2020-online-8th-january-evening-slot |
uxkbzvwhQhZu358eew7k9k2k5ki4pum | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\left| {\overrightarrow a } \right| = \sqrt 3 $$,
$$\left| {\overrightarrow b } \right| = 5,\overrightarrow b .\overrightarrow c = 10$$ and the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$
is ... | [] | null | 30 | Given $$\left| {\overrightarrow a } \right| = \sqrt 3 $$,
$$\left| {\overrightarrow b } \right| = 5$$
<br><br>Given $$\overrightarrow b .\overrightarrow c = 10$$
<br><br>And the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$
is $${\pi \over 3}$$
<br><br>$$ \therefore $$ $$bc\cos {\pi \over 3}$$ = 1... | integer | jee-main-2020-online-9th-january-evening-slot |
I6tuy9NF399qWeyYVhjgy2xukezfmp9m | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let the position vectors of points 'A' and 'B' be
<br/>$$\widehat i + \widehat j + \widehat k$$ and $$2\widehat i + \widehat j + 3\widehat k$$, respectively. A point
'P' divides the line segment AB internally in the
ratio
$$\lambda $$ : 1 (
$$\lambda $$ > 0). If O is the origin and
<br/>$$\overrightarrow {OB} .\ove... | [] | null | 0.8 | Let, $$\overrightarrow a $$ = $$\widehat i + \widehat j + \widehat k$$
<br><br>and $$\overrightarrow b $$ = $$2\widehat i + \widehat j + 3\widehat k$$
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265164/exam_images/hoezbwyywsjaggdoi7yh.webp"><source media="(... | integer | jee-main-2020-online-2nd-september-evening-slot |
XDKiZVSmcBrvn7Fb1njgy2xukfal01ga | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If $$\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k$$, then the value of<br/><br/>
$${\left| {\widehat i \times \left( {\overrightarrow a \times \widehat i} \right)} \right|^2} + {\left| {\widehat j \times \left( {\overrightarrow a \times \widehat j} \right)} \right|^2} + {\left| {\widehat k \times \lef... | [] | null | 18 | Let $$\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$$<br><br>Now $$\widehat i \times \left( {\overrightarrow a \times \widehat i} \right) = \left( {\widehat i.\widehat i} \right)\overrightarrow a - \left( {\widehat i.\overrightarrow a } \right)\widehat i$$<br><br>= $$y\widehat j + z\widehat k$$<br><br>... | integer | jee-main-2020-online-4th-september-evening-slot |
RkTo4epEnfRC9GpYlw1kls5v75w | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k$$ be three given vectors. If $$\overrightarrow r $$ is a vector such that $$\overrightarrow r \times \overrightarrow a = \overrightarrow ... | [] | null | 12 | <p>Given, $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$,</p>
<p>$$\overrightarrow b = \widehat i - \widehat j$$,</p>
<p>$$\overrightarrow c = \widehat i - \widehat j - \widehat k$$</p>
<p>$$\overrightarrow r \times \overrightarrow a = \overrightarrow c \times \overrightarrow a $$</p>
<p>$$ \Righta... | integer | jee-main-2021-online-25th-february-morning-slot |
s7RlE6savu4SIIyjxr1klta1k13 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$ and $$\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ is $$8\sqrt 3 $$ square units, then ... | [] | null | 2 | $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$<br><br>$$\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$$<br><br>Area of parallelogram = $$\left| {\overrightarrow a \times \overrightarrow b } \right|$$<br><br>$$ = \left| {(\widehat i + \alpha \widehat j + 3\widehat k) \times (3... | integer | jee-main-2021-online-25th-february-evening-slot |
hkfGcy1FV2nIdZCA6X1kmiw7pzg | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a $$ = $$\widehat i$$ + 2$$\widehat j$$ $$-$$ 3$$\widehat k$$ and $$\overrightarrow b = 2\widehat i$$ $$-$$ 3$$\widehat j$$ + 5$$\widehat k$$. If $$\overrightarrow r $$ $$\times$$ $$\overrightarrow a $$ = $$\overrightarrow b $$ $$\times$$ $$\overrightarrow r $$, <br/><br/>$$\overrightarrow r $$ .... | [{"identifier": "A", "content": "13"}, {"identifier": "B", "content": "11"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "15"}] | ["D"] | null | Given $$\overrightarrow r $$ $$\times$$ $$\overrightarrow a $$ = $$\overrightarrow b $$ $$\times$$ $$\overrightarrow r $$
<br><br>$$ \Rightarrow $$ $$\overrightarrow r \times \overrightarrow a = - \overrightarrow r \times \overrightarrow b $$<br><br>$$\overrightarrow r \times (\overrightarrow a + \overrightarrow... | mcq | jee-main-2021-online-16th-march-evening-shift |
Y2uW5HKoyGplfqL1nL1kmm2snqv | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two non-zero vectors perpendicular to each other and $$|\overrightarrow a | = |\overrightarrow b |$$. If $$|\overrightarrow a \times \overrightarrow b | = |\overrightarrow a |$$, then the angle between the vectors $$\left( {\overrightarrow a + \overrightarrow b... | [{"identifier": "A", "content": "$${\\sin ^{ - 1}}\\left( {{1 \\over {\\sqrt 6 }}} \\right)$$"}, {"identifier": "B", "content": "$${\\cos ^{ - 1}}\\left( {{1 \\over {\\sqrt 2 }}} \\right)$$"}, {"identifier": "C", "content": "$${\\sin ^{ - 1}}\\left( {{1 \\over {\\sqrt 3 }}} \\right)$$"}, {"identifier": "D", "content": ... | ["D"] | null | $$\overrightarrow a $$ is perpendicular to $$\overrightarrow b $$<br><br>$$ \therefore $$ $$\overrightarrow a $$ . $$\overrightarrow b $$ = 0<br><br>Given, | $$\overrightarrow a $$ $$\times$$ $$\overrightarrow b $$ | = | $$\overrightarrow a $$ |<br><br>and | $$\overrightarrow a $$ | = | $$\overrightarrow b $$ |<br><br>... | mcq | jee-main-2021-online-18th-march-evening-shift |
1krq0xpcz | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If the shortest distance between the lines $$\overrightarrow {{r_1}} = \alpha \widehat i + 2\widehat j + 2\widehat k + \lambda (\widehat i - 2\widehat j + 2\widehat k)$$, $$\lambda$$ $$\in$$ R, $$\alpha$$ > 0 and $$\overrightarrow {{r_2}} = - 4\widehat i - \widehat k + \mu (3\widehat i - 2\widehat j - 2\widehat k... | [] | null | 6 | If $$\overrightarrow r = \overrightarrow a + \lambda \overrightarrow b $$ and $$\overrightarrow r = \overrightarrow c + \lambda \overrightarrow d $$ then shortest distance between two lines is <br><br>$$L = {{(\overrightarrow a - \overrightarrow c ).(\overrightarrow b \times \overrightarrow d )} \over {|b \times ... | integer | jee-main-2021-online-20th-july-morning-shift |
1krw2kdmg | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$ be two vectors. If a vector $$\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \widehat k)$$ is perpendicular to each of the vectors ($$(\overrightarrow p + \overrig... | [] | null | 3 | $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ (Given )<br><br>$$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$<br><br>Now, $$(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q ) = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k}... | integer | jee-main-2021-online-25th-july-morning-shift |
1krzmy5aw | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | If $$\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5$$ and $$\left| {\overrightarrow a \times \overrightarrow b } \right|$$ = 8, then $$\left| {\overrightarrow a .\,\overrightarrow b } \right|$$ is equal to : | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "5"}] | ["A"] | null | $$\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5$$<br><br>$$\left| {\overrightarrow a \times \overrightarrow b } \right| = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\sin \theta = \pm 8$$<br><br>$$\sin \theta = \pm \,{4 \over 5}$$<br><br>$$\therefore$$ $$... | mcq | jee-main-2021-online-25th-july-evening-shift |
1ks05h77r | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$ and $$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$. Then the vector product $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightar... | [{"identifier": "A", "content": "$$5(34\\widehat i - 5\\widehat j + 3\\widehat k)$$"}, {"identifier": "B", "content": "$$7(34\\widehat i - 5\\widehat j + 3\\widehat k)$$"}, {"identifier": "C", "content": "$$7(30\\widehat i - 5\\widehat j + 7\\widehat k)$$"}, {"identifier": "D", "content": "$$5(30\\widehat i - 5\\wideha... | ["B"] | null | $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$<br><br>$$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$<br><br>$$\overrightarrow a + \overrightarrow b = 3\widehat j + 5\widehat k;\overrightarrow a.\overrightarrow b = - 1 + 2 + 6 = 7$$<br><br>$$\left( {\left( {\overrightarrow a \tim... | mcq | jee-main-2021-online-27th-july-morning-shift |
1ks0bqpuy | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b $$ and $$\overrightarrow c = \widehat j - \widehat k$$ be three vectors such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$ and $$\overrightarrow a \,.\,\overrightarrow b = 1$$. If the length of projection ve... | [] | null | 2 | $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$<br><br>Take Dot with $$\overrightarrow c $$<br><br>$$\left( {\overrightarrow a \times \overrightarrow b } \right).\,\overrightarrow c = {\left| {\overrightarrow c } \right|^2} = 2$$<br><br>Projection of $$\overrightarrow b $$ or $$\overrightarrow a... | integer | jee-main-2021-online-27th-july-morning-shift |
1kteolpch | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$$, $$\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k$$ and $$\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k$$ be three vectors such that, $$\left| {\overrightarrow b \times \overrightarrow c } \right| = 5\sqrt 3 $$... | [] | null | 90 | Since, $$\overrightarrow a .\,\overrightarrow b = 0$$<br><br>$$1 + 15 + \alpha \beta = 0 \Rightarrow \alpha \beta = - 16$$ .... (1)<br><br>Also, <br><br>$${\left| {\overrightarrow b \, \times \overrightarrow c } \right|^2} = 75 \Rightarrow (10 + {\beta ^2})14 - {(5 - 3\beta )^2} = 75$$<br><br>$$\Rightarrow$$ 5$$\be... | integer | jee-main-2021-online-27th-august-morning-shift |
1ktoc2aax | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$. Let a vector $$\overrightarrow v $$ be in the plane containing $$\overrightarrow a $$ and $$\overrightarrow b $$. If $$\overrightarrow v $$ is perpendicular to the vector $$3\widehat i... | [] | null | 1494 | $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$<br><br>$$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$<br><br>$$\overrightarrow c = 3\widehat i + 2\widehat j - \widehat k$$<br><br>$$\overrightarrow v = x\overrightarrow a + y\overrightarrow b $$<br><br>$$\overrightarrow v \left( {3\wide... | integer | jee-main-2021-online-1st-september-evening-shift |
1l544t6p5 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$$, $$\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$$ and $$\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k$$ where $$\alpha ,\,\beta \in R$$, be three vectors. If the projection of $$\overrightarrow a $$ on $$\ov... | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "6"}] | ["A"] | null | <p>$$\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$$</p>
<p>$$\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$$</p>
<p>$$\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k$$</p>
<p>Projection of $$\overrightarrow a $$ on $$\overrightarrow c $$ is</p>
<p>$${{\overrightarrow... | mcq | jee-main-2022-online-29th-june-morning-shift |
1l54tqgwd | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$$, $$\overrightarrow b = \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow c $$ be a vector such that $$\overrightarrow a + \left( {\overrightarrow b \times \overrightarrow c } \right) = \overrightarrow 0 $$ and $$\overrig... | [] | null | BONUS | $$
\vec{a} \cdot \vec{b}=(\hat{i}-2 \hat{j}+3 \hat{k}) \cdot(\hat{i}+\hat{j}+\hat{k})=2
$$ ........(i)
<br/><br/>Given: $\vec{a}+(\vec{b} \times \vec{c})=0$
<br/><br/>$$
\Rightarrow \vec{a} \cdot \vec{b}=0
$$ ........(ii)
<br/><br/>Equation (i) and equation (ii) are contradicting. | integer | jee-main-2022-online-29th-june-evening-shift |
1l55iksny | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$$ and $$\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k$$, where $$\alpha \in R$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ is $$... | [{"identifier": "A", "content": "10"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "14"}] | ["D"] | null | <p>$$\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$$ and $$\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k$$</p>
<p>$$\therefore$$ $$\overrightarrow a \times \overrightarrow b = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
\alpha & 2 & { - 1} \cr
... | mcq | jee-main-2022-online-28th-june-evening-shift |
1l55ithsn | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a $$ be a vector which is perpendicular to the vector $$3\widehat i + {1 \over 2}\widehat j + 2\widehat k$$. If $$\overrightarrow a \times \left( {2\widehat i + \widehat k} \right) = 2\widehat i - 13\widehat j - 4\widehat k$$, then the projection of the vector $$\overrightarrow a $$ on the vec... | [{"identifier": "A", "content": "$${1 \\over 3}$$"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "$${5 \\over 3}$$"}, {"identifier": "D", "content": "$${7 \\over 3}$$"}] | ["C"] | null | <p>Let $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$$</p>
<p>and $$\overrightarrow a \,.\,\left( {3\widehat i - {1 \over 2}\widehat j + 2\widehat k} \right) = 0 \Rightarrow 3{a_1} + {{{a_2}} \over 2} + 2{a_3} = 0$$ ..... (i)</p>
<p>and $$\overrightarrow a \times (2\widehat i + \widehat k)... | mcq | jee-main-2022-online-28th-june-evening-shift |
1l567mmea | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>If $$\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 3\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow c = {c_1}\widehat i + {c_2}\widehat j + {c_3}\widehat k$$ are coplanar vectors and $$\overrightarrow a \,.\,\overrightarrow c = 5$$, $$\overrightarrow b \bot \o... | [] | null | 150 | <p>$$2{C_1} + {C_2} + 3{C_3} = 5$$ ...... (i)</p>
<p>$$3{C_1} + 3{C_2} + {C_3} = 0$$ ...... (ii)</p>
<p>$$\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right] = \left| {\matrix{
2 & 1 & 3 \cr
3 & 3 & 1 \cr
{{C_1}} & {{C_2}} & {{C_3}} \cr
} } \right|$$</p>
<p>$$ = 2(3{C_3} - {C_2}) - ... | integer | jee-main-2022-online-28th-june-morning-shift |
1l56rew3i | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be the vectors along the diagonals of a parallelogram having area $$2\sqrt 2 $$. Let the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ be acute, $$|\overrightarrow a | = 1$$, and $$|\overrightarrow a \,.\,\overrightarrow b | = |\overrightarrow a... | [{"identifier": "A", "content": "$${\\pi \\over 4}$$"}, {"identifier": "B", "content": "$$-$$ $${\\pi \\over 4}$$"}, {"identifier": "C", "content": "$${{5\\pi } \\over 6}$$"}, {"identifier": "D", "content": "$${{3\\pi } \\over 4}$$"}] | ["D"] | null | <p>$$\because$$ $$\overrightarrow a $$ and $$\overrightarrow b $$ be the vectors along the diagonals of a parallelogram having area 2$$\sqrt2$$.</p>
<p>$$\therefore$$ $${1 \over 2}|\overrightarrow a \times \overrightarrow b | = 2\sqrt 2 $$</p>
<p>$$|\overrightarrow a ||\overrightarrow b |\sin \theta = 4\sqrt 2 $$</p>... | mcq | jee-main-2022-online-27th-june-evening-shift |
1l57okaze | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$ and $$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$. Then the number of vectors $$\overrightarrow b $$ such that $$\overrightarrow b \times \overrightarrow c = \overrightarrow a $$ and $$|\overrightarrow b | \in $$ {1, 2, ........, ... | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["A"] | null | <p>$$\overrightarrow a = \widehat i + \widehat j - \widehat k$$</p>
<p>$$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$</p>
<p>Now, $$\overrightarrow b \times \overrightarrow c = \overrightarrow a $$</p>
<p>$$\overrightarrow c \,.\,(\overrightarrow b \times \overrightarrow c ) = \overrightarrow c \,... | mcq | jee-main-2022-online-27th-june-morning-shift |
1l59ldqt3 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k$$, $$\lambda$$ $$\in$$ R. If $$\overrightarrow a $$ is a vector such that $$\overrightarrow a \times \overrightarrow b = 13\widehat i - \widehat j - 4\widehat k$$ and $$\overrightarrow a \,.\,\overrightarrow b + 21 = 0$$, then $$\left( {\over... | [] | null | 14 | <p>Let $$\overrightarrow a = x\widehat i = y\widehat j + z\widehat k$$</p>
<p>So, $$\left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
x & y & z \cr
1 & 1 & \lambda \cr
} } \right| = \widehat i(\lambda y - z) + \widehat j(z - \lambda x) + \widehat k(x - y)$$</p>
<p>$$ \Rightarrow \lambda... | integer | jee-main-2022-online-25th-june-evening-shift |
1l5ajnmz4 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\theta$$ be the angle between the vectors $$\overrightarrow a $$ and $$\overrightarrow b $$, where $$|\overrightarrow a | = 4,$$ $$|\overrightarrow b | = 3$$ and $$\theta \in \left( {{\pi \over 4},{\pi \over 3}} \right)$$. Then $${\left| {\left( {\overrightarrow a - \overrightarrow b } \right) \times \left... | [] | null | 576 | <p>$${\left| {\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)} \right|^2} + 4{\left( {\overrightarrow a \,.\,\overrightarrow b } \right)^2}$$</p>
<p>$$ \Rightarrow {\left| {\overrightarrow a \times \overrightarrow a + \overrightarrow a \times \... | integer | jee-main-2022-online-25th-june-morning-shift |
1l5ban31s | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\widehat a$$ and $$\widehat b$$ be two unit vectors such that $$|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$$. If $$\theta$$ $$\in$$ (0, $$\pi$$) is the angle between $$\widehat a$$ and $$\widehat b$$, then among the statements :</p>
<p>(S1) : $$2|\widehat a \times \widehat b| = |\widehat... | [{"identifier": "A", "content": "Only (S1) is true."}, {"identifier": "B", "content": "Only (S2) is true."}, {"identifier": "C", "content": "Both (S1) and (S2) are true."}, {"identifier": "D", "content": "Both (S1) and (S2) are false."}] | ["C"] | null | <p>$$\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right| = 2,\,\theta \in (0,\,\pi )$$</p>
<p>$$ \Rightarrow {\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right|^2} = 4$$</p>
<p>$$ \Rightarrow {\left| {\widehat a} \right|^2} + {\left| {\widehat b} \right|^2} + 4{\left| {\w... | mcq | jee-main-2022-online-24th-june-evening-shift |
1l5c1or9b | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\widehat a$$, $$\widehat b$$ be unit vectors. If $$\overrightarrow c $$ be a vector such that the angle between $$\widehat a$$ and $$\overrightarrow c $$ is $${\pi \over {12}}$$, and $$\widehat b = \overrightarrow c + 2\left( {\overrightarrow c \times \widehat a} \right)$$, then $${\left| {6\overrightarrow ... | [{"identifier": "A", "content": "$$6\\left( {3 - \\sqrt 3 } \\right)$$"}, {"identifier": "B", "content": "$$3 + \\sqrt 3 $$"}, {"identifier": "C", "content": "$$6\\left( {3 + \\sqrt 3 } \\right)$$"}, {"identifier": "D", "content": "$$6\\left( {\\sqrt 3 + 1} \\right)$$"}] | ["C"] | null | $\because \quad \hat{b}=\vec{c}+2(\vec{c} \times \hat{a})$
<br/><br/>
$$
\begin{aligned}
&\Rightarrow \hat{b} \cdot \vec{c}=|\vec{c}|^{2} \\\\
&\therefore \hat{b}-\vec{c}=2(\vec{c} \times \vec{a})
\end{aligned}
$$
<br/><br/>
$$
\begin{aligned}
&\Rightarrow|\hat{b}|^{2}+|\vec{c}|^{2}-2 \hat{b} \cdot \vec{c}=4|\vec{c}|... | mcq | jee-main-2022-online-24th-june-morning-shift |
1l6dwjgp3 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\mathrm{ABC}$$ be a triangle such that $$\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{c}},|\overrightarrow{\mathrm{a}}|=6 \sqrt{2},|\overrightarrow{\mathrm{b}}|=2 \sqrt{3}$$ and $$\vec{b}... | [{"identifier": "A", "content": "both (S1) and (S2) are true"}, {"identifier": "B", "content": "only (S1) is true"}, {"identifier": "C", "content": "only (S2) is true"}, {"identifier": "D", "content": "both (S1) and (S2) are false"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l97sapyl/779dd912-c32e-40a3-8f62-d0f26f1743e1/b6acf9d0-4b5b-11ed-bfde-e1cb3fafe700/file-1l97sapym.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l97sapyl/779dd912-c32e-40a3-8f62-d0f26f1743e1/b6acf9d0-4b5b-11ed-bfde-e1cb3fafe700/fi... | mcq | jee-main-2022-online-25th-july-morning-shift |
1l6f2wm5r | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$$ and let $$\vec{b}$$ be a vector such that $$\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$$ and $$\vec{a} \cdot \vec{b}=3$$. Then the projection of $$\vec{b}$$ on the vector $$\vec{a}-\vec{b}$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{2}{\\sqrt{21}}$$"}, {"identifier": "B", "content": "$$2 \\sqrt{\\frac{3}{7}}$$"}, {"identifier": "C", "content": "$$\n\\frac{2}{3} \\sqrt{\\frac{7}{3}}\n$$"}, {"identifier": "D", "content": "$$\\frac{2}{3}$$"}] | ["A"] | null | <p>$$\overrightarrow a = \widehat i - \widehat j + 2\widehat k$$</p>
<p>$$\overrightarrow a \times \overrightarrow b = 2\widehat i - \widehat k$$</p>
<p>$$\overrightarrow a \,.\,\overrightarrow b = 3$$</p>
<p>$$|\overrightarrow a \times \overrightarrow b {|^2} + |\overrightarrow a \,.\,\overrightarrow b {|^2} = |\... | mcq | jee-main-2022-online-25th-july-evening-shift |
1l6giq99g | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$$. If the projection of $$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$$ on the vector $$-\hat{i}+2 \hat{j}-2 \hat{k}$$ is 30, then $$\alpha$$ is equal ... | [{"identifier": "A", "content": "$$\\frac{15}{2}$$"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "$$\\frac{13}{2}$$"}, {"identifier": "D", "content": "7"}] | ["D"] | null | <p>Given : $$\overrightarrow a = (\alpha ,1, - 1)$$ and $$\overrightarrow b = (2,1, - \alpha )$$</p>
<p>$$\overrightarrow c = \overrightarrow a \times \overrightarrow b = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
\alpha & 1 & { - 1} \cr
2 & 1 & { - \alpha } \cr
} } \right|$$... | mcq | jee-main-2022-online-26th-july-morning-shift |
1l6jc2mfl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$$ and $$\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$$ be two vectors, such that $$\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}$$. Then the projection of $$\vec{b}-2 \vec{a}$$ on $$\vec{b}+\vec{a}$$ is equal to :</p> | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$\\frac{39}{5}$$"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "$$\\frac{46}{5}$$"}] | ["D"] | null | <p>$$\overrightarrow a = \alpha \widehat i + \widehat j + \beta \widehat k$$, $$\overrightarrow b = 3\widehat i - 5\widehat j + 4\widehat k$$</p>
<p>$$\overrightarrow a \times \overrightarrow b = - \widehat i + 9\widehat j + 12\widehat k$$</p>
<p>$$\left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \c... | mcq | jee-main-2022-online-27th-july-morning-shift |
1l6km5v7z | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three non-coplanar vectors such that $$\overrightarrow a $$ $$\times$$ $$\overrightarrow b $$ = 4$$\overrightarrow c $$, $$\overrightarrow b $$ $$\times$$ $$\overrightarrow c $$ = 9$$\overrightarrow a $$ and $$\overrightarrow c $$ $$\times... | [] | null | 36 | <p>Given,</p>
<p>$$\overrightarrow a \times \overrightarrow b = 4\,.\,\overrightarrow c $$ ..... (i)</p>
<p>$$\overrightarrow b \times \overrightarrow c = 9\,.\,\overrightarrow a $$ ..... (ii)</p>
<p>$$\overrightarrow c \times \overrightarrow a = \alpha \,.\,\overrightarrow b $$ .... (iii)</p>
<p>Taking dot produ... | integer | jee-main-2022-online-27th-july-evening-shift |
1l6m5gnnw | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let a vector $$\vec{a}$$ has magnitude 9. Let a vector $$\vec{b}$$ be such that for every $$(x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}$$, the vector $$(x \vec{a}+y \vec{b})$$ is perpendicular to the vector $$(6 y \vec{a}-18 x \vec{b})$$. Then the value of $$|\vec{a} \times \vec{b}|$$ is equal to :</p> | [{"identifier": "A", "content": "$$9 \\sqrt{3}$$"}, {"identifier": "B", "content": "$$27 \\sqrt{3}$$"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "81"}] | ["B"] | null | <p>$$\left( {x\overrightarrow a + y\overrightarrow b } \right).\left( {6y\overrightarrow a - 18x\overrightarrow b } \right) = 0$$</p>
<p>$$ \Rightarrow \left( {6xy|\overrightarrow a {|^2} - 18xy|\overrightarrow b {|^2}} \right) + \left( {6{y^2} - 18{x^2}} \right)\overrightarrow a .\overrightarrow b = 0$$</p>
<p>As g... | mcq | jee-main-2022-online-28th-july-morning-shift |
1l6p2fvoj | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{4}$$. If $$\theta$$ is the angle between the vectors $$(\hat{a}+\hat{b})$$ and $$(\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))$$, then the value of $$164 \,\cos ^{2} \theta$$ is equal to :</p> | [{"identifier": "A", "content": "$$90+27 \\sqrt{2}$$"}, {"identifier": "B", "content": "$$45+18 \\sqrt{2}$$"}, {"identifier": "C", "content": "$$90+3 \\sqrt{2}$$"}, {"identifier": "D", "content": "$$54+90 \\sqrt{2}$$"}] | ["A"] | null | <p>$$\widehat a\,.\,\widehat b = {1 \over {\sqrt 2 }}$$ and $$|\widehat a \times \widehat b| = {1 \over {\sqrt 2 }}$$</p>
<p>$${{\left( {\widehat a + \widehat b} \right)\,.\,\left( {\widehat a + 2\widehat b + 2\left( {\widehat a \times \widehat b} \right)} \right)} \over {\left| {\widehat a + \widehat b} \right|\left| ... | mcq | jee-main-2022-online-29th-july-morning-shift |
1l6rf9wxz | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}, \vec{b}, \vec{c}$$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and
$$(\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})+(\vec{b} \times \vec{c}) \cdot(\vec{c} \times \vec{a})+(\vec{c} \times \vec{a}) \cdot(\vec{a} ... | [{"identifier": "A", "content": "10"}, {"identifier": "B", "content": "14"}, {"identifier": "C", "content": "16"}, {"identifier": "D", "content": "18"}] | ["C"] | null | $|\vec{a}||\vec{b}||\vec{c}|=14$
<br/><br/>
$$
\begin{aligned}
& \vec{a} \wedge \vec{b}=\vec{b} \wedge \vec{c}=\vec{c} \wedge \vec{a}=\theta=\frac{2 \pi}{3} \\\\
& \vec{a} \cdot \vec{b}=-\frac{1}{2}|\vec{a}||\vec{b}| \\\\
& \vec{b} \cdot \vec{c}=-\frac{1}{2}|\vec{b}||\vec{c}| \\\\
& \vec{c} \cdot \vec{a}=-\frac{1}{2}|\... | mcq | jee-main-2022-online-29th-july-evening-shift |
1l6rg17yk | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3$$ and $$|\vec{a} \times \vec{b}|^{2}=75$$. Then $$|\vec{a}|^{2}$$ is equal to __________.</p> | [] | null | 14 | $\because|\vec{a}+\dot{b}|^{2}=|\vec{a}|^{2}+2|b|^{2}$
<br/><br/>or $|\vec{a}|^{2}+|\vec{b}|^{2}+2 \vec{a} \cdot \vec{b}=|\vec{a}|^{2}+2|\vec{b}|^{2}$
<br/><br/>$\therefore|\vec{b}|^{2}=6$
<br/><br/>Now $|\vec{a} \times \vec{b}|^{2}=|\vec{a}|^{2}|\vec{b}|^{2}-(\vec{a} \cdot \vec{b})^{2}$
<br/><br/>$$
75=|\vec{a}|^{... | integer | jee-main-2022-online-29th-july-evening-shift |
ldo7cuo5 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}$ be three vectors. If $\vec{r}$ is a vector such
that, $\vec{r} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a}=0$, then $25|\vec{r}|^{2}$ is equal to : | [{"identifier": "A", "content": "336"}, {"identifier": "B", "content": "449"}, {"identifier": "C", "content": "339"}, {"identifier": "D", "content": "560"}] | ["C"] | null | $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$
<br/><br/>$\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$
<br/><br/>$\overrightarrow{\mathrm{c}}=\hat{5 \mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$
<br/><br/>$(\overrightarrow{\mathrm{r}}-\o... | mcq | jee-main-2023-online-31st-january-evening-shift |
ldoaq5ul | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that
<br/><br/>$|\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$.
<br/><br/>If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2 \pi}{3}$, then $\left(\frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\righ... | [] | null | 3 | $2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$
<br/><br/>$\vec{a} \times(2 \vec{b}+3 \vec{c})=0$
<br/><br/>$$
\begin{aligned}
& \vec{a}=\lambda(2 \vec{b}+3 \vec{c}) \\\\
& |\vec{a}|^{2}=\lambda^{2}\left(4|b|^{2}+9|c|^{2}+12 \vec{b} \cdot \vec{c}\right) \\\\
& 31=31 \lambda^{2} \\\\
& \lambda=\pm 1 \\\\
& \vec{a... | integer | jee-main-2023-online-31st-january-evening-shift |
1ldooe6vv | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>$$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$$ and $$D(4,5,0),|\lambda| \leq 5$$ are the vertices of a quadrilateral $$A B C D$$. If its area is 18 square units, then $$5-6 \lambda$$ is equal to __________.</p> | [] | null | 11 | $$
\begin{aligned}
& \mathrm{A}(2,6,2) \quad \mathrm{B}(-4,0, \lambda), \mathrm{C}(2,3,-1) \mathrm{D}(4,5,0) \\\\
& \text { Area }=\frac{1}{2}|\overrightarrow{B D} \times \overrightarrow{A C}|=18 \\\\
& \overrightarrow{A C} \times \overrightarrow{B D}=\left|\begin{array}{ccc}
\hat{i} & j & k \\\\
0 & -3 & -3 \\\\
8 & 5... | integer | jee-main-2023-online-1st-february-morning-shift |
1ldptp6vk | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$$ and $$|\vec{a} \times \vec{b}|=\sqrt{48}$$. Then $$(\vec{a} \cdot \vec{b})^{2}$$ is equal to ___________.</p> | [] | null | 36 | $|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$ and $|\vec{a} \times \vec{b}|=\sqrt{48}$
<br/><br/>$$
\begin{aligned}
& \Rightarrow |\vec{a} \times \vec{b}|^{2}+(\vec{a} \cdot \vec{b})^{2}=|\vec{a}|^{2}|\vec{b}|^{2} \\\\
& \Rightarrow 48+(\vec{a} \cdot \vec{b})^{2}=6 \times 14 \\\\
& \Rightarrow (\vec{a} \cdot \vec{b})^{2}=8... | integer | jee-main-2023-online-31st-january-morning-shift |
ldqx819b | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $\vec{a}$ and $\vec{b}$ be two vectors, Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$, then the value of $\vec{b} \cdot \vec{c}$ is : | [{"identifier": "A", "content": "$-48$"}, {"identifier": "B", "content": "$-60$"}, {"identifier": "C", "content": "$-84$"}, {"identifier": "D", "content": "$-24$"}] | ["A"] | null | <p>$$\overrightarrow b .\overrightarrow c = \overrightarrow b .(2\overline a \times \overrightarrow b ) - 3\overline b .\overrightarrow b $$</p>
<p>$$ = 0 - 3|\overline b {|^2} = - 48$$</p> | mcq | jee-main-2023-online-30th-january-evening-shift |
1ldr7kwam | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let a unit vector $$\widehat{O P}$$ make angles $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes $$\mathrm{OX}$$, $$\mathrm{OY}, \mathrm{OZ}$$ respectively, where $$\beta \in\left(0, \frac{\pi}{2}\right)$$. If $$\widehat{\mathrm{OP}}$$ is perpendicular to the plane through points $$(1,2... | [{"identifier": "A", "content": "$$\\alpha \\in\\left(\\frac{\\pi}{2}, \\pi\\right)$$ and $$\\gamma \\in\\left(\\frac{\\pi}{2}, \\pi\\right)$$"}, {"identifier": "B", "content": "$$\\alpha \\in\\left(0, \\frac{\\pi}{2}\\right)$$ and $$\\gamma \\in\\left(\\frac{\\pi}{2}, \\pi\\right)$$"}, {"identifier": "C", "content": "... | ["A"] | null | <p>Let $$A \equiv (1,2,3),B \equiv (2,3,4),C \equiv (1,5,7)$$</p>
<p>$$\overrightarrow n = \overrightarrow {AB} \times \overrightarrow {AC} = \left| {\matrix{
i & j & k \cr
1 & 1 & 1 \cr
0 & 3 & 4 \cr
} } \right|$$</p>
<p>$$ = \widehat i - 4\widehat j + 3\widehat k$$</p>
<p>$$\widehat {OP} = {{ \pm ... | mcq | jee-main-2023-online-30th-january-morning-shift |
1ldseppqy | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat k,\overrightarrow r \times \overrightarrow b + \overrightarrow b \times \overrightarrow c = \overrightarrow 0 $$ and $$\overrightarrow r \,.\,\ov... | [{"identifier": "A", "content": "36"}, {"identifier": "B", "content": "30"}, {"identifier": "C", "content": "34"}, {"identifier": "D", "content": "32"}] | ["C"] | null | <p>$$(\overrightarrow r - \overrightarrow c ) \times \overrightarrow b = 0$$</p>
<p>$$\overrightarrow r = \lambda \overrightarrow b + \overrightarrow c $$</p>
<p>$$ \Rightarrow \lambda \overrightarrow b \,.\,\overrightarrow a + \overrightarrow c \,.\,\overrightarrow a = 0$$</p>
<p>$$ \Rightarrow \lambda (3) + (7 ... | mcq | jee-main-2023-online-29th-january-evening-shift |
1ldsfbgwo | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a = 4\widehat i + 3\widehat j$$ and $$\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$$. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightarrow b } \right) + 25 = 0,\overrightarrow c \,.(\widehat i + \widehat j + \wid... | [{"identifier": "A", "content": "$$\\frac{3}{\\sqrt2}$$"}, {"identifier": "B", "content": "$$\\frac{1}{\\sqrt2}$$"}, {"identifier": "C", "content": "$$\\frac{1}{5}$$"}, {"identifier": "D", "content": "$$\\frac{5}{\\sqrt2}$$"}] | ["D"] | null | <p>$$[\matrix{
{\overrightarrow c } & {\overrightarrow a } & {\overrightarrow b } \cr
} ] = - 25$$</p>
<p>Let $$\overrightarrow c = l\widehat i + n\widehat j + n\widehat k$$</p>
<p>$$\left| {\matrix{
l & m & n \cr
4 & 3 & 0 \cr
3 & { - 4} & 5 \cr
} } \right| = - 25$$</p>
<p>$$ \Rightarrow 3l ... | mcq | jee-main-2023-online-29th-january-evening-shift |
1ldwxj5lf | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow a = \widehat i + 2\widehat j + \lambda \widehat k,\overrightarrow b = 3\widehat i - 5\widehat j - \lambda \widehat k,\overrightarrow a \,.\,\overrightarrow c = 7,2\overrightarrow b \,.\,\overrightarrow c + 43 = 0,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \over... | [] | null | 8 | $$
\begin{aligned}
& \vec{a}=\hat{i}+2 \hat{j}+\lambda \hat{k}, \vec{b}=3 \hat{i}-5 \hat{j}-\lambda \hat{k}, \vec{a} \cdot \vec{c}=7 \\\\
& \vec{a} \times \vec{c}-\vec{b} \times \vec{c}=\overrightarrow{0} \\\\
& (\vec{a}-\vec{b}) \times \vec{c}=0 \Rightarrow(\vec{a}-\vec{b}) \text { is paralleled to } \vec{c} \\\\
& \v... | integer | jee-main-2023-online-24th-january-evening-shift |
1lgow35tq | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$|\vec{a}|=2,|\vec{b}|=3$$ and the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$ be $$\frac{\pi}{4}$$. Then $$|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^{2}$$ is equal to :</p> | [{"identifier": "A", "content": "441"}, {"identifier": "B", "content": "482"}, {"identifier": "C", "content": "841"}, {"identifier": "D", "content": "882"}] | ["D"] | null | $$
\begin{aligned}
& |\vec{a}|=2 \\\\
& |\vec{b}|=3 \\\\
& \vec{a} \cdot \vec{b}=\frac{\pi}{4}
\end{aligned}
$$
<br/><br/>$$
\begin{aligned}
& |(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^2 \\\\
& = |-3 \vec{a} \times \vec{b}+4 \vec{b} \times \vec{a}|^2 \\\\
& = |-3 \vec{a} \times \vec{b}-4 \vec{a} \times \vec{b}|... | mcq | jee-main-2023-online-13th-april-evening-shift |
1lgoxv6q0 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let for a triangle $$\mathrm{ABC}$$,</p>
<p>$$\overrightarrow{\mathrm{AB}}=-2 \hat{i}+\hat{j}+3 \hat{k}$$</p>
<p>$$\overrightarrow{\mathrm{CB}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$$</p>
<p>$$\overrightarrow{\mathrm{CA}}=4 \hat{i}+3 \hat{j}+\delta \hat{k}$$</p>
<p>If $$\delta > 0$$ and the area of the tria... | [{"identifier": "A", "content": "60"}, {"identifier": "B", "content": "54"}, {"identifier": "C", "content": "120"}, {"identifier": "D", "content": "108"}] | ["A"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lh1phdby/e74db975-74e1-4987-80c4-b8ae4380dc69/bb54dde0-e665-11ed-9cfb-9bcc868d407f/file-1lh1phdbz.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lh1phdby/e74db975-74e1-4987-80c4-b8ae4380dc69/bb54dde0-e665-11ed-9cfb-9bcc868d407f/fi... | mcq | jee-main-2023-online-13th-april-evening-shift |
1lgpxqwcg | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$$ and $$\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$$. If a vector $$\vec{d}$$ satisfies $$\vec{d} \times \vec{b}=\vec{c} \times \vec{b}$$ and $$\vec{d} \cdot \vec{a}=24$$, then $$|\vec{d}|^{2}$$ is equal to :</p> | [{"identifier": "A", "content": "313"}, {"identifier": "B", "content": "413"}, {"identifier": "C", "content": "423"}, {"identifier": "D", "content": "323"}] | ["B"] | null | Given that $$\vec{d} \times \vec{b} = \vec{c} \times \vec{b}$$, we can rewrite this as:
<br/><br/>$$(\vec{d} - \vec{c}) \times \vec{b} = \vec{0}$$
<br/><br/>This implies that the vector $$\vec{d} - \vec{c}$$ is a scalar multiple of $$\vec{b}$$:
<br/><br/>$$\vec{d} = \vec{c} + \lambda \vec{b}$$
<br/><br/>Also, we ar... | mcq | jee-main-2023-online-13th-april-morning-shift |
1lgq10ehr | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$$ and $$\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$$. If $$\vec{b}$$ is a vector such that $$\vec{a}=\vec{b} \times \vec{c}$$ and $$|\vec{b}|^{2}=50$$, then $$|72-| \vec{b}+\left.\vec{c}\right|^{2} \mid$$ is equal to __________.</p> | [] | null | 66 | <p>Given that $$\vec{a} = \vec{b} \times \vec{c}$$, we can find the magnitudes of $$\vec{a}$$ and $$\vec{c}$$:</p>
<p>$$|\vec{a}| = \sqrt{3^2 + 1^2 + (-1)^2} = \sqrt{11}$$
<br/><br/>$$|\vec{c}| = \sqrt{2^2 + (-3)^2 + 3^2} = \sqrt{22}$$</p>
<p>We know that the magnitude of the cross product of two vectors is equal to th... | integer | jee-main-2023-online-13th-april-morning-shift |
1lgsw25ll | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$$ and $$\vec{b}=\hat{i}+\hat{j}-\hat{k}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c}=11,
\vec{b} \cdot(\vec{a} \times \vec{c})=27$$ and $$\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|$$, then $$|\vec{a} \times \vec{c}|^{2}$$ is equal to _________.</p> | [] | null | 285 | Given,
<br/><br/>$$
\begin{aligned}
& \vec{a}=\hat{i}+2 \hat{j}+3 \hat{k} \\\\
& \vec{b}=\hat{i}+\hat{j}-\hat{k} \\\\
& \vec{a} \cdot \vec{c}=11 \\\\
& \vec{b} \cdot(\vec{a} \times \vec{c})=27 \\\\
& \vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}| \\\\
& (\vec{b} \times \vec{a}) \cdot \vec{c}=27
\end{aligned}
$$
<br/><br/>$$... | integer | jee-main-2023-online-11th-april-evening-shift |
1lguvx5sl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}$$ be a non-zero vector parallel to the line of intersection of the two planes described by $$\hat{i}+\hat{j}, \hat{i}+\hat{k}$$ and $$\hat{i}-\hat{j}, \hat{j}-\hat{k}$$. If $$\theta$$ is the angle between the vector $$\vec{a}$$ and the vector $$\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}$$ and $$\vec{a} \cdot ... | [{"identifier": "A", "content": "$$\\left(\\frac{\\pi}{3}, 3 \\sqrt{6}\\right)$$"}, {"identifier": "B", "content": "$$\\left(\\frac{\\pi}{3}, 6\\right)$$"}, {"identifier": "C", "content": "$$\\left(\\frac{\\pi}{4}, 3 \\sqrt{6}\\right)$$"}, {"identifier": "D", "content": "$$\\left(\\frac{\\pi}{4}, 6\\right)$$"}] | ["D"] | null | We have, $$\vec{a}$$ is non-zero vector parallel to the line of intersection of the two planes described by $\hat{\mathbf{i}}+\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}$.
<br/><br/>Let $\mathbf{n}_1$ and $\mathbf{n}_2$ are the normal ve... | mcq | jee-main-2023-online-11th-april-morning-shift |
1lgvq0gag | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$$ and $$\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$$. Let $$\vec{d}$$ be a vector which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$, and $$\vec{c} \cdot \vec{d}=12$$. Then $$(-\hat{i}+\hat{j}-\hat{k}) \cdot(\vec{c} \times \vec{d})$$ is equal to :... | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "42"}, {"identifier": "C", "content": "44"}, {"identifier": "D", "content": "48"}] | ["C"] | null | If $\vec{d}$ is $\perp$ to both $\vec{a}$ and $\vec{b}$ then
<br/><br/>$$
\vec{d}=\lambda(\vec{a} \times \vec{b})=\lambda\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
2 & 7 & -1 \\
3 & 0 & 5
\end{array}\right|=(35 \hat{i}-13 \hat{j}-21 \hat{k}) \lambda
$$
<br/><br/>$$
\begin{aligned}
& \text { but } \vec{c} \... | mcq | jee-main-2023-online-10th-april-evening-shift |
1lgxh78lc | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let O be the origin and the position vector of the point P be $$ - \widehat i - 2\widehat j + 3\widehat k$$. If the position vectors of the points A, B and C are $$ - 2\widehat i + \widehat j - 3\widehat k,2\widehat i + 4\widehat j - 2\widehat k$$ and $$ - 4\widehat i + 2\widehat j - \widehat k$$ respectively, then ... | [{"identifier": "A", "content": "$$\\frac{7}{3}$$"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "$$\\frac{10}{3}$$"}, {"identifier": "D", "content": "$$\\frac{8}{3}$$"}] | ["B"] | null | Given, the position vector of point P is :
$ \overrightarrow{OP} = -\widehat{i} - 2\widehat{j} + 3\widehat{k} $
<br/><br/>Position vectors of points A, B, and C are :
<br/><br/>$ \overrightarrow{OA} = -2\widehat{i} + \widehat{j} - 3\widehat{k} $
<br/><br/>$ \overrightarrow{OB} = 2\widehat{i} + 4\widehat{j} - 2\widehat... | mcq | jee-main-2023-online-10th-april-morning-shift |
1lgylle5f | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>The area of the quadrilateral $$\mathrm{ABCD}$$ with vertices $$\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)$$ and $$\mathrm{D}(1,-6,-7)$$ is equal to :</p> | [{"identifier": "A", "content": "48"}, {"identifier": "B", "content": "$$8 \\sqrt{38}$$"}, {"identifier": "C", "content": "54"}, {"identifier": "D", "content": "$$9 \\sqrt{38}$$"}] | ["B"] | null | $$
\begin{aligned}
& \text { Here } \overrightarrow{\mathrm{AC}}=(-2-2) \hat{i}+(-3-1) \hat{j}+(5-1) \hat{k} \\\\
& =-4 \hat{i}-4 \hat{j}+4 \hat{k} \\\\
& \overrightarrow{\mathrm{BD}}=(1-1) \hat{i}+(-6-2) \hat{j}+(-7-5) \hat{k} \\\\
& =-8 \hat{j}-12 \hat{k}
\end{aligned}
$$
<br/><br/>So, area of quadrilateral $=\frac{1... | mcq | jee-main-2023-online-8th-april-evening-shift |
1lh00gxyv | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}$$ and $$\vec{c}$$ be vectors such that $$\vec{a} \times \vec{c}=\vec{a} \times \vec{b}$$. If <br/><br/>$$\vec{a} \cdot \vec{c}=-12, \vec{c} \cdot(\hat{i}-2 \hat{j}+\hat{k})=5$$, then $$\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})$$ i... | [] | null | 11 | Let $\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$
<br/><br/>Now, $\vec{a} \cdot \vec{c}=-12$
<br/><br/>$$
\Rightarrow 6 c_1+9 c_2+12 c_3=-12
$$ ..............(i)
<br/><br/>Also, $\vec{c} \cdot(\hat{i}-2 \hat{j}+\hat{k})=5$
<br/><br/>$$
\Rightarrow c_1-2 c_2+c_3=5
$$ ................(ii)
<br/><br/>$$
\begin{aligned}
& \... | integer | jee-main-2023-online-8th-april-morning-shift |
1lh21kev1 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$$ and $$\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$$.
If $$\vec{d}$$ is a vector perpendicular to both $$\vec{b}$$ and $$\vec{c}$$, and $$\vec{a} \cdot \vec{d}=18$$, then $$|\vec{a} \times \vec{d}|^{2}$$ is equal to :</p> | [{"identifier": "A", "content": "680"}, {"identifier": "B", "content": "720"}, {"identifier": "C", "content": "760"}, {"identifier": "D", "content": "640"}] | ["B"] | null | <p>Given vectors :
<br/><br/>$ \vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} $
<br/><br/>$ \vec{b} = \hat{i} - 2\hat{j} - 2\hat{k} $
<br/><br/>$ \vec{c} = -\hat{i} + 4\hat{j} + 3\hat{k} $</p>
<p>Since $ \vec{d} $ is perpendicular to both $ \vec{b} $ and $ \vec{c} $, its direction is given by their cross product :</p>
<p>$ \... | mcq | jee-main-2023-online-6th-april-morning-shift |
lsan8kau | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $\overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=-\hat{i}-8 \hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}=4 \hat{i}+\mathrm{c}_2 \hat{j}+\mathrm{c}_3 \hat{k}$ be three vectors such that $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}}... | [] | null | 38 | $\begin{aligned} & \vec{a}=\hat{i}+\hat{j}+k \\\\ & \vec{b}=\hat{i}+8 \hat{j}+2 k \\\\ & \vec{c}=4 \hat{i}+c_2 \hat{j}+c_3 k \\\\ & \vec{b} \times \vec{a}=\vec{c} \times \vec{a} \\\\ & (\vec{b}-\vec{c}) \times \vec{a}=0 \\\\ & \vec{b}-\vec{c}=\lambda \vec{\alpha} \\\\ & \vec{b}=\vec{c}+\lambda \vec{\alpha}\end{aligned}... | integer | jee-main-2024-online-1st-february-evening-shift |
lsaojnb9 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | Let $\overrightarrow{\mathrm{a}}=-5 \hat{i}+\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}-4 \hat{k}$ and
<br/><br/>$\overrightarrow{\mathrm{c}}=(((\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \hat{i}) \times \hat{i}) \times \hat{i}$. Then $\vec{c} \cdot(-\hat{i}+\hat{j}+\ha... | [{"identifier": "A", "content": "-12"}, {"identifier": "B", "content": "-10"}, {"identifier": "C", "content": "-13"}, {"identifier": "D", "content": "-15"}] | ["A"] | null | $\begin{aligned} & \vec{a}=-5 \cdot \hat{i}+\hat{j}-3 \hat{k}, \vec{b}=\hat{i}+2 \hat{j}-4 \hat{k} \\\\ & \vec{c}=(((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i} \\\\ & =(((\vec{a} \cdot \hat{i}) \vec{b}-(\vec{b} \cdot \hat{i}) \vec{a}) \times \hat{i}) \times \hat{i} \\\\ & =((-5 \vec{b}-\vec{... | mcq | jee-main-2024-online-1st-february-morning-shift |
jaoe38c1lscmwyjl | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let the position vectors of the vertices $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ of a triangle be $$2 \hat{i}+2 \hat{j}+\hat{k}, \hat{i}+2 \hat{j}+2 \hat{k}$$ and $$2 \hat{i}+\hat{j}+2 \hat{k}$$ respectively. Let $$l_1, l_2$$ and $$l_3$$ be the lengths of perpendiculars drawn from the ortho center of the trian... | [{"identifier": "A", "content": "$$\\frac{1}{4}$$"}, {"identifier": "B", "content": "$$\\frac{1}{5}$$"}, {"identifier": "C", "content": "$$\\frac{1}{3}$$"}, {"identifier": "D", "content": "$$\\frac{1}{2}$$"}] | ["D"] | null | <p>$$\triangle \mathrm{ABC}$$ is equilateral</p>
<p>Orthocentre and centroid will be same</p>
<p>$$\mathrm{G}\left(\frac{5}{3}, \frac{5}{3}, \frac{5}{3}\right)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1vv91f/efa66592-d248-4f32-8e31-2416eaa514c4/4caa5d30-d411-11ee-b9d5-0585032231f0/fil... | mcq | jee-main-2024-online-27th-january-evening-shift |
jaoe38c1lsd4wwd2 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}$$ and $$(\vec{a}-\vec{b}+\hat{i}) \cdot \vec{c}=-3$$. Then $$|\vec{c}|^2$$ is equal to ________.</p> | [] | null | 38 | <p>$$\begin{aligned}
& (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})+24 \hat{\mathrm{j}}-6 \hat{\mathrm{k}} \\
& (5 \hat{\mathrm{i}}+\hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \times \overrightarrow{\mathrm{c}}=2(... | integer | jee-main-2024-online-31st-january-evening-shift |
jaoe38c1lse565zc | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}$$ and $$\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}$$ be three vectors. If a vectors $$\vec{p}$$ satisfies $$\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$$ and $$\vec{p} \cdot \vec{a}=0$$, then $$\vec{p} \cdot(\hat{i}-\hat{j}-\hat{k})$$ is equal... | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "32"}, {"identifier": "C", "content": "36"}, {"identifier": "D", "content": "28"}] | ["B"] | null | <p>$$\begin{aligned}
& \overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0} \\
& (\overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}=\overrightarrow{0} \\
& \overrightarrow{\mathrm{p}}-\o... | mcq | jee-main-2024-online-31st-january-morning-shift |
jaoe38c1lse59x61 | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>The distance of the point $$Q(0,2,-2)$$ form the line passing through the point $$P(5,-4, 3)$$ and perpendicular to the lines $$\vec{r}=(-3 \hat{i}+2 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+5 \hat{k}), \lambda \in \mathbb{R}$$ and $$\vec{r}=(\hat{i}-2 \hat{j}+\hat{k})+\mu(-\hat{i}+3 \hat{j}+2 \hat{k}), \mu \in \mathbb{... | [{"identifier": "A", "content": "$$\\sqrt{74}$$\n"}, {"identifier": "B", "content": "$$\\sqrt{86}$$\n"}, {"identifier": "C", "content": "$$\\sqrt{54}$$\n"}, {"identifier": "D", "content": "$$\\sqrt{20}$$"}] | ["A"] | null | <p>A vector in the direction of the required line can be obtained by cross product of</p>
<p>$$\begin{aligned}
& \left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
2 & 3 & 5 \\
-1 & 3 & 2
\end{array}\right| \\\\
& =-9 \hat{i}-9 \hat{j}+9 \hat{k}
\end{aligned}$$</p>
<p>Required line<... | mcq | jee-main-2024-online-31st-january-morning-shift |
jaoe38c1lse5sysr | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}|=1,|\vec{b}|=4$$, and $$\vec{a} \cdot \vec{b}=2$$. If $$\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$$ and the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\alpha$$, then $$192 \sin ^2 \alpha$$ is equal to ________.</p> | [] | null | 48 | <p>$$\begin{aligned}
& \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=(2 \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{b}}-3|\mathrm{b}|^2 \\
& |\mathrm{~b}||c| \cos \alpha=-3|\mathrm{~b}|^2 \\
& |\mathrm{c}| \cos \alpha=-12 \text {, as }|\mathrm{b}|=4 \\
&... | integer | jee-main-2024-online-31st-january-morning-shift |
jaoe38c1lsfkro4t | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}$$, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $$\mathrm{{{area\,of\,the\,quadrilateral\,OA\,BC} \over {area\,of\,S}}}$$ is equal to _________.</p> | [{"identifier": "A", "content": "7"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "10"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsr8xnpy/836457b4-b43d-438c-b5ce-783fa67cdc3d/c728ff60-ce37-11ee-9412-cd4f9c6f2c40/file-6y3zli1lsr8xnpz.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsr8xnpy/836457b4-b43d-438c-b5ce-783fa67cdc3d/c728ff60-ce37-11ee... | mcq | jee-main-2024-online-29th-january-evening-shift |
1lsg4dytc | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}$$. Let a vector $$\vec{b}$$ be such that the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$\frac{\pi}{4}$$ and $$|\vec{b}|^2=6$$. If $$\vec{a} \cdot \vec{b}=3 \sqrt{2}$$, then the value of $$\left(\alpha^2+\beta^2\right)|\vec{a} \times... | [{"identifier": "A", "content": "85"}, {"identifier": "B", "content": "90"}, {"identifier": "C", "content": "75"}, {"identifier": "D", "content": "95"}] | ["B"] | null | <p>$$\begin{aligned}
& |\overrightarrow{\mathrm{b}}|^2=6 ;|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos \theta=3 \sqrt{2} \\
& |\overrightarrow{\mathrm{a}}|^2|\overrightarrow{\mathrm{b}}|^2 \cos ^2 \theta=18 \\
& |\overrightarrow{\mathrm{a}}|^2=6
\end{aligned}$$</p>
<p>Also $$1+\alpha^2+\beta^2=6$$</p... | mcq | jee-main-2024-online-30th-january-evening-shift |
1lsg4jz4o | maths | vector-algebra | vector-or-cross-product-of-two-vectors-and-its-applications | <p>Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{b}|=1$$ and $$|\vec{b} \times \vec{a}|=2$$. Then $$|(\vec{b} \times \vec{a})-\vec{b}|^2$$ is equal to</p> | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "4"}] | ["C"] | null | <p>To find the value of $$|(\vec{b} \times \vec{a})-\vec{b}|^2$$, we can use properties of vector operations and magnitudes. Given $$|\vec{b}| = 1$$ and $$|\vec{b} \times \vec{a}| = 2$$, let's break down the calculation step by step:</p>
<p>Firstly, we observe that the cross product of two vectors $$\vec{b} \times \vec... | mcq | jee-main-2024-online-30th-january-evening-shift |
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