question stringlengths 79 9.83k | answer stringlengths 33 9.39k |
|---|---|
Let a, b and c be distinct positive numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i+\widehat k$$ and $$c\widehat i + c\widehat j + b\widehat k$$ are co-planar, then c is equal to :
Options:
[{"identifier": "A", "content": "$${2 \\over {{1 \\over a} + {1 \\over b}}}$$"}, {"identifier": "B",... | ["D"]
Explanation:
Because vectors are coplanar<br><br>Hence, $$\left| {\matrix{
a & a & c \cr
1 & 0 & 1 \cr
c & c & b \cr
} } \right| = 0$$<br><br>$$ \Rightarrow {c^2} = ab \Rightarrow c = \sqrt {ab} $$ |
Let A, B, C be three points whose position vectors respectively are
<p>$$\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k$$</p>
<p>$$\overrightarrow b = 2\widehat i + \alpha \widehat j + 4\widehat k,\,\alpha \in R$$</p>
<p>$$\overrightarrow c = 3\widehat i - 2\widehat j + 5\widehat k$$</p>
<p>If $$\alpha$... | ["A"]
Explanation:
$\overrightarrow{A B} \| \overrightarrow{A C}$ if
<br/><br/>
$\frac{1}{2}=\frac{\alpha-4}{-6}=\frac{1}{2}$
<br/><br/>
$\Rightarrow \alpha=1$
<br/><br/>
$\vec{a}, \vec{b}, \vec{c}$ are non-collinear for $\alpha=2$ (smallest positive integer)
<br/><br/>
Mid point of $B C=M\left(\frac{5}{2}, 0, \frac{9... |
<p>Let the vectors $$\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, \vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}$$ and $$\vec{c}=t \hat{i}-t \hat{j}+\hat{k}, t \in \mathbf{R}$$ be such that for $$\alpha, \beta, \gamma \in \mathbf{R}, \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\overrightarrow{0} \Rightarrow \alpha=\beta... | ["C"]
Explanation:
<p>Clearly $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ are non-coplanar</p>
<p>$$\left| {\matrix{
{1 + t} & {1 - t} & 1 \cr
{1 - t} & {1 + t} & 2 \cr
t & { - t} & 1 \cr
} } \right| \ne 0$$</p>
<p>$$ \Rightarrow (1 + t)(1 + t + 2t) - (1 - t)(1 - t - 2t) + 1... |
<p>Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that <br/><br/>$${{QA} \over {AR}} = {{RB} \over {BP}} = {{PC} \over {CQ}} = {1 \over 2}$$. Then $${{Area(\Delta PQR)} \over {Area(\Delta ABC)}}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "$$\\frac{5}{2}$... | ["D"]
Explanation:
Let the position vector of $P, Q, R$ be $\overrightarrow{0}, \overrightarrow{a}, \overrightarrow{b}$
<br><br>$\Rightarrow$ Position vector of $A=\frac{2 \overrightarrow{a}+\overrightarrow{b}}{3}, $
<br><br>Position vector of $B=\frac{2 \overrightarrow{b}}{3}$ and
<br><br>Position vector of $C=\fra... |
Let $\mathrm{ABCD}$ be a quadrilateral. If $\mathrm{E}$ and $\mathrm{F}$ are the mid points of the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ respectively and $(\overrightarrow{A B}-\overrightarrow{B C})+(\overrightarrow{A D}-\overrightarrow{D C})=k \overrightarrow{F E}$, then $k$ is equal to :
Options:
[{"identifier":... | ["C"]
Explanation:
<p>Let the position vectors of $A, B, C,$ and $D$ be $\vec{a}, \vec{b}, \vec{c},$ and $\vec{d}$, respectively.</p>
<p>Then the position vector of $E$ is:</p>
<p>$$
\vec{E} = \frac{\vec{a} + \vec{c}}{2}
$$</p>
<p>And the position vector of $F$ is:</p>
<p>$$
\vec{F} = \frac{\vec{b} + \vec{d}}{2}
$$</p... |
<p>For any vector $$\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$$, with $$10\left|a_{i}\right|<1, i=1,2,3$$, consider the following statements :</p>
<p>(A): $$\max \left\{\left|a_{1}\right|,\left|a_{2}\right|,\left|a_{3}\right|\right\} \leq|\vec{a}|$$</p>
<p>(B) : $$|\vec{a}| \leq 3 \max \left\{\left|a_{1}\rig... | ["D"]
Explanation:
We have,
<br/><br/>$$
\begin{aligned}
& 10\left|a_i\right|<1, i=1,2,3 \\\\
& \text { Let } \left|a_1\right| \geq\left|a_2\right| \geq\left|a_3\right| \\\\
& |\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2} \geq \sqrt{a_1^2} \\\\
& \therefore|\vec{a}| \geq\left|a_1\right| \text { or } \max \left\{\left|a_1\right|,... |
<p>If the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ are respectively the circumcenter and the orthocentre of a $$\triangle \mathrm{ABC}$$, then $$\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PC}}$$ is
equal to :</p>
Options:
[{"identifier": "A", "content": "$$\\overrightarrow {QP} $... | ["B"]
Explanation:
1. **Circumcenter $ P $**:
<br/><br/>The circumcenter of a triangle is equidistant from the vertices of the triangle. It is the center of the circumcircle, the circle that passes through all three vertices of the triangle.
<br/><br/>2. **Orthocenter $ Q $**:
<br/><br/>The orthocenter of a triangl... |
<p>An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $$\overrightarrow {OP} = \overrightarrow u ,\overrightarrow {OR} = \overrightarrow v $$, and $$\overrightarrow {OQ} = \alpha \overrightarrow u + \beta \overrightarrow v $$, then $$\alpha ,{\beta ^2}$$ are the roots... | ["D"]
Explanation:
An arc $P Q$ of a circle subtends a right angle at its centre $O$. The mid-point of an $\operatorname{arc} P Q$ is $R$. So, $P R=R Q$
<br><br>Also given that, $$\overrightarrow {OP} = \overrightarrow u ,\overrightarrow {OR} = \overrightarrow v $$, and $$\overrightarrow {OQ} = \alpha \overrightar... |
<p>If the points with position vectors $$\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}$$ are collinear, then $$(19 \alpha-6 \beta)^{2}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "49"}, {"identif... | ["C"]
Explanation:
Given : Points with position vectors
<br/><br/>$$
\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}
$$
<br/><br/>and $\frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}$ are collinear.
<br/><br/>So, $\frac{\alpha-6}{6-\frac{9}{2}}=\frac{10-11}{11-\beta}=\frac{13-11}{11+8}$
<br/><br/>$$
... |
<p>The position vectors of the vertices $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ of a triangle are $$2 \hat{i}-3 \hat{j}+3 \hat{k}, 2 \hat{i}+2 \hat{j}+3 \hat{k}$$ and $$-\hat{i}+\hat{j}+3 \hat{k}$$ respectively. Let $$l$$ denotes the length of the angle bisector $$\mathrm{AD}$$ of $$\angle \mathrm{BAC}$$ where $$... | ["A"]
Explanation:
<p>$$\begin{aligned}
& \mathrm{AB}=5 \\
& \mathrm{AC}=5
\end{aligned}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1v5c9e/0c2c210a-ab17-46c0-813d-6d826d2adf6f/7c1dc320-d40e-11ee-b9d5-0585032231f0/file-1lt1v5c9f.png?format=png" data-orsrc="https://app-content.cd... |
<p>Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero vectors such that $$\vec{b}$$ and $$\vec{c}$$ are non-collinear. If $$\vec{a}+5 \vec{b}$$ is collinear with $$\vec{c}, \vec{b}+6 \vec{c}$$ is collinear with $$\vec{a}$$ and $$\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}$$, then $$\alpha+\beta$$ is... | ["D"]
Explanation:
<p>$$\begin{aligned}
& \vec{a}+5 \vec{b}=\lambda \vec{c} \\
& \vec{b}+6 \vec{c}=\mu \vec{a}
\end{aligned}$$</p>
<p>Eliminating $$\vec{a}$$</p>
<p>$$\begin{aligned}
& \lambda \overrightarrow{\mathrm{c}}-5 \overrightarrow{\mathrm{b}}=\frac{6}{\mu} \overrightarrow{\mathrm{c}}+\frac{1}{\mu} \overrightar... |
If $$\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c $$ are vectors such that $$\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right] = 4$$ then $$\left[ {\overrightarrow a \, \times \overrightarrow b \,\,\overrightarrow b \times \,\overrightarrow c \,\,\overrightarrow c \... | ["A"]
Explanation:
We have, $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\overrightarrow b \times \overrightarrow c \,\,\overrightarrow c \times \overrightarrow a } \right]$$
<br><br>$$ = \left( {\overrightarrow a \times \overrightarrow b } \right).\,\,\left\{ {\left( {\overrightarrow b \times \overri... |
If $$\overrightarrow u \,,\overrightarrow v $$ and $$\overrightarrow w $$ are three non-coplanar vectors, then $$\,\left( {\overrightarrow u + \overrightarrow v - \overrightarrow w } \right).\left( {\overrightarrow u - \overrightarrow v } \right) \times \left( {\overrightarrow v - \overrightarrow w} \right)$$ equal... | ["C"]
Explanation:
$$\left( {\overrightarrow u + \overrightarrow v - \overrightarrow w } \right).\left( {\overrightarrow u \times \overrightarrow v - \overrightarrow u \times \overrightarrow w - \overrightarrow v \times \overrightarrow v + \overrightarrow v \times \overrightarrow w } \right)$$
<br><br>$$ = \l... |
If $${\overrightarrow a ,\overrightarrow b ,\overrightarrow c }$$ are non-coplanar vectors and $$\lambda $$ is a real number, then the vectors $${\overrightarrow a + 2\overrightarrow b + 3\overrightarrow c ,\,\,\lambda \overrightarrow b + 4\overrightarrow c }$$ and $$\left( {2\lambda - 1} \right)\overrightarrow c $... | ["C"]
Explanation:
Vectors $$\overrightarrow a + 2\overrightarrow b + 3\overrightarrow c ,\lambda \overrightarrow b + 4\overrightarrow c ,\,\,\,$$ <br><br>and $$\left( {2\lambda - 1} \right)\overrightarrow c $$ are
<br><br>coplanar if $$\left| {\matrix{
1 & 2 & 3 \cr
0 & \lambda & 4 \cr ... |
Let $$\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \widehat j\,\,\, + \,\,\,\left( {1 - x} \right)\widehat k$$ and $$\overrightarrow c \,\, = \,\,y\widehat i + x\widehat j + \left( {1 + x - y} \right)\widehat k.$$ Then $$\left[ {\overrightarrow a ,\overri... | ["D"]
Explanation:
$$\overrightarrow a = \widehat j - \widehat k,\overrightarrow b = x\widehat i + \overrightarrow j + \left( {1 - x} \right)\widehat k$$
<br><br>and $$\overrightarrow c = y\widehat i + x\widehat j + \left( {1 + x - y} \right)\widehat k$$
<br><br>$$\left[ {\overrightarrow a \,\overrightarrow b \,\... |
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are non coplanar vectors and $$\lambda $$ is a real number then <br/><br/>$$\left[ {\lambda \left( {\overrightarrow a + \overrightarrow b } \right)\,\,\,\,\,\,\,\,{\lambda ^2}\overrightarrow b \,\,\,\,\,\,\,\,\lambda \overrightarrow c } \right] = \left[ {... | ["B"]
Explanation:
$$\left[ {\lambda \left( {\overrightarrow a + \overrightarrow b } \right){\lambda ^2}\overrightarrow b \,\,\,\lambda \overrightarrow c } \right] = \left[ {\overrightarrow a \,\,\overrightarrow b + \overrightarrow c \,\,\overrightarrow b } \right]$$
<br><br>$$ \Rightarrow {\lambda ^4}\left[ {\overr... |
If $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)$$ where $${\overrightarrow a ,\overrightarrow b }$$ and $${\overrightarrow c }$$ are any three vectors such that $$\overrightarrow a .\ov... | ["D"]
Explanation:
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c $$
<br><br>$$\,\,\,\,\,\,\,\,\,\,\,\,\, = \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right),\overrightarrow a .\overrightarrow b \ne 0,\,\,\overrightarrow b .\overrightarro... |
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow c = x\widehat i + \left( {x - 2} \right)\widehat j - \widehat k\,\,.$$ If the vectors $$\overrightarrow c $$ lies in the plane of $$\overrightarrow a $$ and $$\overrightar... | ["B"]
Explanation:
Given $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat k$$
<br><br>and $$\overrightarrow c = x\widehat i + \left( {x - 2} \right)\widehat j - \widehat k$$
<br><br>If $$\overrightarrow c $$ lies in the plane of $$\overrightarrow a ... |
The vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k$$ lies in the plane of the vectors
<br/>$$\overrightarrow b = \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat j + \widehat k$$ and bisects the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$.Then which ... | ["D"]
Explanation:
As $$\overrightarrow a $$ lies in the plane of $$\overrightarrow b $$ and $$\overrightarrow c $$
<br><br>$$\therefore$$ $$\overrightarrow a = \overrightarrow b + \lambda \overrightarrow c $$
<br><br>$$ \Rightarrow \alpha \widehat i + 2\widehat j + \beta \widehat k = \widehat i + \widehat j + \lamb... |
If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ are non-coplanar vectors and $$p,q$$ are real numbers, then the equality $$\left[ {3\overrightarrow u \,\,p\overrightarrow v \,\,p\overrightarrow w } \right] - \left[ {p\overrightarrow v \,\,\overrightarrow w \,\,q\overrightarrow u } \right] - \left[ {2\ov... | ["D"]
Explanation:
$$\left[ {3\overrightarrow u \,\,p\overrightarrow v \,\,p\overrightarrow \omega } \right] - \left[ {p\overrightarrow v \,\,\overrightarrow \omega \,\,q\overrightarrow u } \right] - \left[ {2\overrightarrow \omega \,\,q\overrightarrow v \,\,q\overrightarrow u } \right] = 0$$
<br><br>$$ \Rightarrow... |
Let $$\overrightarrow a = \widehat j - \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k.$$ Then the vector $$\overrightarrow b $$ satisfying $$\overrightarrow a \times \overrightarrow b + \overrightarrow c = \overrightarrow 0 $$ and $$\overrightarrow a .\overrightarrow b = 3$$ :
Option... | ["D"]
Explanation:
$$\overrightarrow c = \overrightarrow b \times \overrightarrow a $$
<br><br>$$ \Rightarrow \overrightarrow b .\overrightarrow c = \overrightarrow b .\left( {\overrightarrow b \times \overrightarrow a } \right) \Rightarrow \overrightarrow b .\overrightarrow c = 0$$
<br><br>$$ \Rightarrow \left( ... |
If $$\overrightarrow a = {1 \over {\sqrt {10} }}\left( {3\widehat i + \widehat k} \right)$$ and $$\overrightarrow b = {1 \over 7}\left( {2\widehat i + 3\widehat j - 6\widehat k} \right),$$ then the value
<br/><br>of $$\left( {2\overrightarrow a - \overrightarrow b } \right)\left[ {\left( {\overrightarrow a \times ... | ["D"]
Explanation:
We have $$\overrightarrow a .\overrightarrow b = 0,\,\,\overrightarrow a .\overrightarrow a = 1,\,\,\overrightarrow b .\overrightarrow b = 1$$
<br><br>$$\left( {2\overrightarrow a - \overrightarrow b } \right).\left[ {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\... |
If $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\,\,\overrightarrow b \times \overrightarrow c \,\,\,\,\overrightarrow c \times \overrightarrow a } \right] = \lambda {\left[ {\overrightarrow a\,\,\,\,\,\,\,\, \overrightarrow b \,\,\,\,\,\,\,\,\overrightarrow c } \right]^2}$$ then $$\lambda $$ is equal to... | ["B"]
Explanation:
$$L.H.S$$ $$ = \left( {\overrightarrow a \times \overrightarrow b } \right).\left[ {\left( {\overrightarrow b \times \overrightarrow c } \right) \times \left( {\overrightarrow c \times \overrightarrow a } \right)} \right]$$
<br><br>$$ = \left( {\overrightarrow a \times \overrightarrow b } \right... |
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero vectors such that no two of them are collinear and <br/><br/>$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \r... | ["C"]
Explanation:
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\overrightarrow a $$
<br><br>$$ \Rightarrow - \overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } ... |
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = {{\sqrt 3 } \over 2}\left( {\overrightarrow b + \overrightarrow c } \right).$$ If $${\overrightarrow b }$$ is not parallel... | ["B"]
Explanation:
$$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = {{\sqrt 3 } \over 2}\left( {\overrightarrow b + \overrightarrow c } \right)$$
<br><br>$$ \Rightarrow \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow a .\ove... |
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors, out of which vectors $$\overrightarrow b $$ and $$\overrightarrow c $$ are non-parallel. If $$\alpha $$ and $$\beta $$ are the angles which vector $$\overrightarrow a $$ makes with vectors $$\overrightarrow b $$ and $$... | ["B"]
Explanation:
$$\left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow a .\overrightarrow b } \right).\overrightarrow c = {1 \over 2}\overrightarrow b $$
<br><br>$$ \because $$ $$\overrightarrow b \,\,$$ & $$\overrightarrow c \,\,$$ are linearly indepe... |
Let $$\alpha $$ $$ \in $$ R and the three vectors <br/><br/>$$\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 2\widehat i + \widehat j - \alpha \widehat k$$ <br/><br/>and $$\overrightarrow c = \alpha \widehat i - 2\widehat j + 3\widehat k$$. <br/><br/>Then the set
S = {$$\al... | ["D"]
Explanation:
Since these vectors are coplanar then,<br><br>
$$\left| {\matrix{
\alpha & 1 & 3 \cr
2 & 1 & { - \alpha } \cr
\alpha & { - 2} & 3 \cr
} } \right| = 0$$<br><br>
Now, $$\alpha (3 - 2\alpha ) - 1\left( {6 + {\alpha ^2}} \right) + 3\left( { - 4 - \alpha } \righ... |
If the volume of parallelopiped formed by the vectors $$\widehat i + \lambda \widehat j + \widehat k$$, $$\widehat j + \lambda \widehat k$$ and $$\lambda \widehat i + \widehat k$$ is minimum, then $$\lambda $$ is
equal to :
Options:
[{"identifier": "A", "content": "$$ - {1 \\over {\\sqrt 3 }}$$"}, {"identifier": "B", ... | ["D"]
Explanation:
$$V = \left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right] = \left| {\matrix{
1 & \lambda & 1 \cr
0 & 1 & \lambda \cr
\lambda & 0 & 1 \cr
} } \right|$$<br><br>
$$ \Rightarrow 1 - \lambda \left( { - {\lambda ^2}} \right) + 1.\left( {0 - \... |
The sum of the distinct real values of $$\mu $$, for which the vectors, $$\mu \widehat i + \widehat j + \widehat k,$$ $$\widehat i + \mu \widehat j + \widehat k,$$ $$\widehat i + \widehat j + \mu \widehat k$$ are co-planar, is :
Options:
[{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$-$$... | ["B"]
Explanation:
$$\left| {\matrix{
\mu & 1 & 1 \cr
1 & \mu & 1 \cr
1 & 1 & \mu \cr
} } \right| = 0$$
<br><br>$$\mu \left( {{\mu ^2} - 1} \right) - 1\left( {\mu - 1} \right) + 1\left( {1 - \mu } \right) = 0$$
<br><br>$${\mu ^3} - \mu - \mu + 1 + 1\mu = 0$$
<br><br>$${\... |
Let $$\overrightarrow a $$ = $$\widehat i - \widehat j$$, $$\overrightarrow b $$ = $$\widehat i + \widehat j + \widehat k$$ and $$\overrightarrow c $$
<br/><br/>be a vector such that $$\overrightarrow a $$ × $$\overrightarrow c $$ + $$\overrightarrow b $$ = $$\overrightarrow 0 $$ <br/><br/>and $$\overrightarrow a $$ .... | ["B"]
Explanation:
Given that,
<br><br>$$\overrightarrow a \times \overrightarrow c + \overrightarrow b = \overrightarrow 0 $$
<br><br>$$ \Rightarrow $$ $$\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow c } \right) + \overrightarrow a \times \overrightarrow b = \overrightar... |
Let the volume of a parallelopiped whose
coterminous edges are given by
<br/><br>$$\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k$$, $$\overrightarrow v = \widehat i + \widehat j + 3\widehat k$$ and
<br/><br>$$\overrightarrow w = 2\widehat i + \widehat j + \widehat k$$ be 1 cu. unit. If $$\theta $$... | ["A"]
Explanation:
Volume of parallelopiped = 1
<br><br>$$\left| {\left[ {\matrix{
{\overrightarrow u } & {\overrightarrow v } & {\overrightarrow w } \cr
} } \right]} \right|$$ = 1
<br><br>$$ \Rightarrow $$ $$\left| {\matrix{
1 & 1 & \lambda \cr
1 & 1 & 3 \cr
2 & 1 &... |
If the vectors, $$\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k$$,
<br/><br>
$$\overrightarrow q = a\widehat i + \left( {a + 1} \right)\widehat j + a\widehat k$$ and
<br/><br>
$$\overrightarrow r = a\widehat i + a\widehat j + \left( {a + 1} \right)\widehat k\left( {a \in R} \right)... | 1
Explanation:
$$ \because $$ $$\overrightarrow p$$, $$\overrightarrow q$$, $$\overrightarrow r$$ are coplanar
<br><br>$$ \therefore $$ $$\left[ {\matrix{
{\overrightarrow p } & {\overrightarrow q } & {\overrightarrow r } \cr
} } \right]$$ = 0
<br><br>$$ \Rightarrow $$ $$\left| {\matrix{
{a + 1} &... |
Let x<sub>0</sub> be the point of Local maxima of $$f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$$, where <br/>$$\overrightarrow a = x\widehat i - 2\widehat j + 3\widehat k$$, $$\overrightarrow b = - 2\widehat i + x\widehat j - \widehat k$$, $$\overrightarrow c = 7\widehat... | ["D"]
Explanation:
$$f(x) = \overrightarrow a \,.\,(\overrightarrow b \times \overrightarrow c )$$<br><br>$$ = \left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$$<br><br>$$ = \left| {\matrix{
x & { - 2} & 3 \cr
{ - 2} & x & { - 1} \cr
7 & { - 2} & x \cr ... |
If the volume of a parallelopiped, whose<br/> coterminus edges are given by the <br/>vectors $$\overrightarrow a = \widehat i + \widehat j + n\widehat k$$, <br/>$$\overrightarrow b = 2\widehat i + 4\widehat j - n\widehat k$$ and <br/>$$\overrightarrow c = \widehat i + n\widehat j + 3\widehat k$$ ($$n \ge 0$$), is 1... | ["B"]
Explanation:
We know, Volume(V) = $$\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]$$
<br><br>$$ \Rightarrow $$ 158 = $$\left| {\matrix{
1 & 1 & n \cr
2 & 4 & { - n} \cr
1 & n & 3 \cr
} } \right|$$
<br><br>$$ \Rightarrow $$ (12 + n<sup>2</sup>) – (... |
Let three vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarrow c $$ is coplanar <br/>with $$\overrightarrow a $$ and $$\overrightarrow b $$,
$$\overrightarrow a .\overrightarrow c $$ = 7 and
$$\overrightarrow b $$ is perpendicular to $$\overrightarrow c $$, ... | 75
Explanation:
$$\overrightarrow c = \lambda (\overrightarrow b \times (\overrightarrow a \times \overrightarrow b ))$$<br><br>$$ = \lambda ((\overrightarrow b \,.\,\overrightarrow b )\overrightarrow a - (\overrightarrow b \,.\,\overrightarrow a )\overrightarrow b )$$<br><br>$$ = \lambda (5( - \widehat i + \wideh... |
If $$\overrightarrow a $$ and $$\overrightarrow b $$ are perpendicular, then <br/>$$\overrightarrow a \times \left( {\overrightarrow a \times \left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right)} \right)$$ is equal to :
Options:
[{"identifier": "A", "content": "$${... | ["D"]
Explanation:
$$\overrightarrow a \,.\,\overrightarrow b = 0$$<br><br>$$\overrightarrow a \times (\overrightarrow a \times \overrightarrow b ) = (\overrightarrow a \,.\,\overrightarrow b )\overrightarrow a - (\overrightarrow a \,.\,\overrightarrow a )\overrightarrow b = - |\overrightarrow a {|^2}\overrighta... |
Let $$\overrightarrow c $$ be a vector perpendicular to the vectors, $$\overrightarrow a $$ = $$\widehat i$$ + $$\widehat j$$ $$-$$ $$\widehat k$$ and <br/>$$\overrightarrow b $$ = $$\widehat i$$ + 2$$\widehat j$$ + $$\widehat k$$. If $$\overrightarrow c \,.\,\left( {\widehat i + \widehat j + 3\widehat k} \right)$$ = 8... | 28
Explanation:
$$\overrightarrow a \times \overrightarrow b = \left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
1 & 1 & { - 1} \cr
1 & 2 & 1 \cr
} } \right| = (3, - 2,1)$$<br><br>$$\overrightarrow c \bot \overrightarrow a ,\overrightarrow c \bot \overrightarr... |
Let $$\overrightarrow a $$ = 2$$\widehat i$$ $$-$$ 3$$\widehat j$$ + 4$$\widehat k$$ and $$\overrightarrow b $$ = 7$$\widehat i$$ + $$\widehat j$$ $$-$$ 6$$\widehat k$$.<br/><br/>If $$\overrightarrow r $$ $$\times$$ $$\overrightarrow a $$ = $$\overrightarrow r $$ $$\times$$ $$\overrightarrow b $$, $$\overrightarrow r $... | ["D"]
Explanation:
$$\overrightarrow a = (2, - 3,4)$$, $$\overrightarrow b = (7,1, - 6)$$<br><br>$$\overrightarrow r \times \overrightarrow a - \overrightarrow r \times \overrightarrow b = 0$$<br><br>$$\overrightarrow r \times (\overrightarrow a - \overrightarrow b ) = 0$$<br><br>$$\overrightarrow r = \lambda... |
If $$\overrightarrow a = \alpha \widehat i + \beta \widehat j + 3\widehat k$$,<br/><br/>$$\overrightarrow b = - \beta \widehat i - \alpha \widehat j - \widehat k$$ and <br/><br/>$$\overrightarrow c = \widehat i - 2\widehat j - \widehat k$$<br/><br/>such that $$\overrightarrow a \,.\,\overrightarrow b = 1$$ and $$\... | 2
Explanation:
$$\overrightarrow a .\overrightarrow b = 1 \Rightarrow - \alpha \beta - \alpha \beta - 3 = 1$$<br><br>$$ \Rightarrow \alpha \beta = - 2$$ .... (i)<br><br>$$\overrightarrow b .\overrightarrow c = - 3 \Rightarrow - \beta + 2\alpha + 1 = - 3$$<br><br>$$2\alpha - \beta = - 4$$ ..... (ii)<br><... |
Let O be the origin. Let $$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k$$ and $$\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$$, x, y$$\in$$R, x > 0, be such that $$\left| {\overrightarrow {PQ} } \right| = \sqrt {20} $$ and the vector $$\overrightarrow {OP} $$ is perpendicular ... | ["B"]
Explanation:
$$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k\,$$
<br/><br/>$$\overrightarrow {OP} \bot \overrightarrow {OQ} $$<br><br>$$\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$$<br><br>$$\overrightarrow {PQ} = \left( { - 1 - x} \right)\widehat i + \left( {2 - y} \rig... |
Let $$\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow a .\,\overrightarrow c = \left| {\overrightarrow c } \right|,\left| {\overrightarrow c - \overrightarrow a } \right| = 2\sqrt 2 $$... | ["D"]
Explanation:
$$\left| {\overrightarrow a } \right| = 3 = a;\overrightarrow a \,.\,\overrightarrow c = c$$<br><br>Now, $$\left| {\overrightarrow c - \overrightarrow a } \right| = 2\sqrt 2 $$<br><br>$$ \Rightarrow {c^2} + {a^2} - 2\overrightarrow c \,.\,\overrightarrow a = 8$$<br><br>$$ \Rightarrow {c^2} + 9 - ... |
Let a vector $${\overrightarrow a }$$ be coplanar with vectors $$\overrightarrow b = 2\widehat i + \widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j + \widehat k$$. If $${\overrightarrow a}$$ is perpendicular to $$\overrightarrow d = 3\widehat i + 2\widehat j + 6\widehat k$$, and $$\left| {... | ["A"]
Explanation:
$$\overrightarrow a = \lambda \overrightarrow b + \mu \overrightarrow c = \widehat i(2\lambda + \mu ) + \widehat j(\lambda - \mu ) + \widehat k(\lambda + \mu )$$<br><br>$$\overrightarrow a \,.\,\overrightarrow d = 0 = 3(2\lambda + \mu ) + 2(\lambda - \mu ) + 6(\lambda + \mu )$$<br><br>$$ \... |
Let three vectors $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$, $$\overrightarrow b \times \overrightarrow c = \overrightarrow a $$ and $$\left| {\overrightarrow a } \right| = 2$$. Then which one of the fol... | ["D"]
Explanation:
(1) $$\overrightarrow a \times \left( {(\overrightarrow b + \overrightarrow c ) \times (\overrightarrow b \times \overrightarrow c )} \right)$$<br><br>$$ = \overrightarrow a ( - \overrightarrow b \times \overrightarrow c + \overrightarrow c \times \overrightarrow b ) = - 2\left( {\overrightar... |
Let the vectors<br/><br/>$$(2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k,(1 + b)\widehat i + 2b\widehat j - b\widehat k$$ and $$(2 + b)\widehat i + 2b\widehat j + (1 - b)\widehat k$$, $$a,b,c, \in R$$<br><br/> be co-planar. Then which of the following is true?</br>
Options:
[{"identifier": "A", "c... | ["A"]
Explanation:
If the vectors are co-planar,<br><br>$$\left| {\matrix{
{a + b + 2} & {a + 2b + c} & { - b - c} \cr
{b + 1} & {2b} & { - b} \cr
{b + 2} & {2b} & {1 - b} \cr
} } \right| = 0$$<br><br>Now, $${R_3} \to {R_3} - {R_2},{R_1} \to {R_1} - {R_2}$$<br><br>So, $$\left| ... |
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three vectors such that $$\overrightarrow a $$ = $$\overrightarrow b $$ $$\times$$ ($$\overrightarrow b $$ $$\times$$ $$\overrightarrow c $$). If magnitudes of the vectors $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarro... | ["B"]
Explanation:
$$\overrightarrow a = \left( {\overrightarrow b .\,\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow b \,.\,\overrightarrow b } \right)\overrightarrow c $$<br><br>$$ = 1.2\cos \theta \overrightarrow b - \overrightarrow c $$<br><br>$$ \Rightarrow \overrightarrow a = 2\cos \th... |
Let $$\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k$$, $$\overrightarrow b = 3\widehat i + \beta \widehat j - \alpha \widehat k$$ and $$\overrightarrow c = -\alpha \widehat i - 2\widehat j + \widehat k$$, where $$\alpha$$ and $$\beta$$ are integers. If $$\overrightarrow a \,.\,\overrightarr... | 9
Explanation:
$$\overrightarrow a = (1, - \alpha ,\beta )$$<br><br>$$\overrightarrow b = (3,\beta , - \alpha )$$<br><br>$$\overrightarrow c = ( - \alpha , - 2,1);\alpha ,\beta \in I$$<br><br>$$\overrightarrow a \,.\,\overrightarrow b = - 1 \Rightarrow 3 - \alpha \beta - \alpha \beta = - 1$$<br><br>$$ \Righta... |
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = \widehat j - \widehat k$$. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow a \times \overrightarrow c = \overrightarrow b $$ and $$\overrightarrow a .\overrightarrow c = 3$$, then $$\overrightarrow a .(\ove... | ["A"]
Explanation:
$$\left| {\overrightarrow a } \right| = \sqrt 3 $$; $$\overrightarrow a .\overrightarrow c = 3$$; $$\overrightarrow a \times \overrightarrow b = - 2\widehat i + \widehat j + \widehat k$$, $$\overrightarrow a \times \overrightarrow c = \overrightarrow b $$<br><br>Cross with $$\overrightarrow a ... |
Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ three vectors mutually perpendicular to each other and have same magnitude. If a vector $${ \overrightarrow r } $$ satisfies.
<br/><br/>$$\overrightarrow a \times \{ (\overrightarrow r - \overrightarrow b ) \times \overrightarrow a \} + \overrightar... | ["C"]
Explanation:
Suppose $$\overrightarrow r = x\overrightarrow a + y\overrightarrow b + 2\overrightarrow c $$<br><br>and $$\left| {\overrightarrow a } \right| = \left| {\overrightarrow b } \right| = \left| {\overrightarrow c } \right| = k$$<br><br>$$\overrightarrow a \times \{ (\overrightarrow r - \overrightar... |
<p>If $$\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2$$ and $$\overrightarrow c \,.\,\overrightarrow a = 3$$, then the value of $$\left[ {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right),\,\overrightarrow b \times \left( {\overrigh... | ["A"]
Explanation:
<p>$$\because$$ $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = 3\overrightarrow b - \overrightarrow c = \overrightarrow u $$</p>
<p>$$\overrightarrow b \times \left( {\overrightarrow c \times \overrightarrow a } \right) = \overrightarrow c - 2\overri... |
<p>Let a vector $$\overrightarrow c $$ be coplanar with the vectors $$\overrightarrow a = - \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = 2\widehat i + \widehat j - \widehat k$$. If the vector $$\overrightarrow c $$ also satisfies the conditions $$\overrightarrow c \,.\,\left[ {\left( {\overrighta... | ["C"]
Explanation:
<p>Given,</p>
<p>$$\overrightarrow a = - \widehat i + \widehat j + \widehat k$$</p>
<p>$$\overrightarrow b = 2\widehat i + \widehat j - \widehat k$$</p>
<p>and let $$\overrightarrow c = x\widehat i + y\widehat j + z\widehat k$$</p>
<p>Now, $$\overrightarrow a + \overrightarrow b = \widehat i +... |
<p>$$
\text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2} \text {, then }|\vec{b} \times 2 \hat{j}|
$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "4"}, {"identifi... | ["B"]
Explanation:
<p>Given, $$\overrightarrow a = 2\widehat i - \widehat j + 5\widehat k$$ and $$\overrightarrow b = \alpha \widehat i + \beta \widehat j + 2\widehat k$$</p>
<p>Also, $$\left( {\left( {\overrightarrow a \times \overrightarrow b } \right) \times i} \right)\,.\,\widehat k = {{23} \over 2}$$</p>
<p>$$... |
<p>Let $$\overrightarrow{\mathrm{a}}=3 \hat{i}+\hat{j}$$ and $$\overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}+\hat{k}$$. Let $$\overrightarrow{\mathrm{c}}$$ be a vector satisfying $$\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\overrightarrow{\mathrm{b}}+\lambda \ove... | ["A"]
Explanation:
<p>$$\overrightarrow a = 3\widehat i + \widehat j$$ & $$\overrightarrow b = \widehat i + 2\widehat j + \widehat k$$</p>
<p>$$\overrightarrow a \times (\overrightarrow b \times \overrightarrow c ) = (\overrightarrow a \,.\,\overrightarrow c )\overrightarrow b - (\overrightarrow a \,.\,\overright... |
<p>Let $$\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}$$ and $$\vec{u}$$ be a vector such that $$|\vec{u}|=\alpha>0$$. If the minimum value of the scalar triple product $$\left[ {\matrix{
{\overrightarrow u } & {\overrightarrow v } & {\overrightarrow w } \cr
} } ... | 3501
Explanation:
$\vec{v} \times \vec{w}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ \alpha & 2 & -3 \\ 2 \alpha & 1 & -1\end{array}\right|=\hat{i}-5 \alpha \hat{j}-3 \alpha \hat{k}$
<br/><br/>$$\left[ {\matrix{
{\overrightarrow u } & {\overrightarrow v } & {\overrightarrow w } \cr
} } \right] = \o... |
Let $\lambda \in \mathbb{R}, \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}$.
<br/><br/>If $((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}$,<br/><br/> then $|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2$ is equal ... | ["B"]
Explanation:
<p>$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right) + \overrightarrow b \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right) \times \left( {\overrightarrow a - \overrightarrow b } \right)$$</p>
<p>$$ = \left( {\overrightarro... |
<p>If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are three non-zero vectors and $$\widehat n$$ is a unit vector perpendicular to $$\overrightarrow c $$ such that $$\overrightarrow a = \alpha \overrightarrow b - \widehat n,(\alpha \ne 0)$$ and $$\overrightarrow b \,.\overrightarrow c = 12$$, then $... | ["D"]
Explanation:
<p>$$\widehat n = \alpha \overrightarrow b - \overrightarrow a $$</p>
<p>$$\overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } \right) = \left( {\overrightarrow c \,.\,\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow c \,.\,\overrightarrow a } \rig... |
<p>Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero non-coplanar vectors. Let the position vectors of four points $$A,B,C$$ and $$D$$ be $$\overrightarrow a - \overrightarrow b + \overrightarrow c ,\lambda \overrightarrow a - 3\overrightarrow b + 4\overrightarrow c , -... | 2
Explanation:
$\overline{A B}=(\lambda-1) \bar{a}-2 \bar{b}+3 \bar{c}$
<br/><br/>
$$
\overline{A C}=2 \bar{a}+3 \bar{b}-4 \bar{c}
$$<br/><br/>$$
\overline{A D}=\bar{a}-3 \bar{b}+5 \bar{c}
$$<br/><br/>$$
\left|\begin{array}{ccc}
\lambda-1 & -2 & 3 \\
-2 & 3 & -4 \\
1 & -3 & 5
\end{array}\right|=0
$$<br/><br/>$$
\Right... |
<p>Let $$\overrightarrow a = - \widehat i - \widehat j + \widehat k,\overrightarrow a \,.\,\overrightarrow b = 1$$ and $$\overrightarrow a \times \overrightarrow b = \widehat i - \widehat j$$. Then $$\overrightarrow a - 6\overrightarrow b $$ is equal to :</p>
Options:
[{"identifier": "A", "content": "$$3\\left( ... | ["A"]
Explanation:
$$
\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})
$$<br/><br/>
Taking cross product with $\vec{a}$<br/><br/>
$$
\begin{aligned}
& \Rightarrow \vec{a} \times(\vec{a} \times \vec{b})=\vec{a} \times(\hat{i}-\hat{j}) \\\\
& \Rightarrow (\vec{a} \cdot ... |
<p>If the four points, whose position vectors are $$3\widehat i - 4\widehat j + 2\widehat k,\widehat i + 2\widehat j - \widehat k, - 2\widehat i - \widehat j + 3\widehat k$$ and $$5\widehat i - 2\alpha \widehat j + 4\widehat k$$ are coplanar, then $$\alpha$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "... | ["A"]
Explanation:
Let $\mathrm{A}:(3,-4,2) \quad \mathrm{C}:(-2,-1,3)$<br/><br/>
$$
\text { B : }(1,2,-1) \quad \text { D: }(5,-2 \alpha, 4)
$$<br/><br/>
A, B, C, D are coplanar points, then<br/><br/>
$$
\begin{aligned}
& \Rightarrow\left|\begin{array}{ccc}
1-3 & 2+4 & -1-2 \\
-2-3 & -1+4 & 3-2 \\
5-3 & -2 \alpha+4 &... |
<p>Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three non zero vectors such that $$\overrightarrow b $$ . $$\overrightarrow c $$ = 0 and $$\overrightarrow a \times (\overrightarrow b \times \overrightarrow c ) = {{\overrightarrow b - \overrightarrow c } \over 2}$$. If $$\overright... | ["C"]
Explanation:
$\vec{b}(\vec{a} \cdot \vec{c})-\vec{c}(\vec{a} \cdot \vec{b})=\frac{\vec{b}-\vec{c}}{2}$ $\vec{a} \cdot \vec{c}=\frac{1}{2}, \quad \vec{a} \cdot \vec{b}=\frac{1}{2}$
<br/><br/>
$$
\begin{aligned}
(\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d}) & =(\vec{b} \cdot \vec{d})(\vec{a} \cdot \vec{c}... |
<p>Let $$\overrightarrow u = \widehat i - \widehat j - 2\widehat k,\overrightarrow v = 2\widehat i + \widehat j - \widehat k,\overrightarrow v .\,\overrightarrow w = 2$$ and $$\overrightarrow v \times \overrightarrow w = \overrightarrow u + \lambda \overrightarrow v $$. Then $$\overrightarrow u .\,\overrightarrow... | ["D"]
Explanation:
$$
\begin{aligned}
&\begin{aligned}
& \vec{v} \times \vec{w}=(\vec{u}+\lambda \vec{v})=\hat{i}-\hat{j}-2 \hat{k}+\lambda(2 \hat{i}+\hat{j}-\hat{k}) \\\\
& =(2 \lambda+1) \hat{i}+(\lambda-1) \hat{j}-(2+\lambda) \hat{k} \\
&
\end{aligned}\\
&\begin{aligned}
& \text { Now, } \vec{v} \cdot(\vec{v} \time... |
Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i}-\hat{j}+\hat{k}, \hat{i}+2 \hat{j}+\mu \hat{k}$ and $3 \hat{i}-4 \hat{j}+5 \hat{k}$, where $\lambda-\mu=5$, are coplanar, then $\sum\limits_{(\lambda, \mu) \in S} 80\left(\lambda^2+\mu^2\right)$ is equal to :
Options:
[{"identifier": "A"... | ["D"]
Explanation:
Step 1: Given condition for coplanarity
<br/><br/>For three vectors to be coplanar, their scalar triple product must be zero. We have the vectors A, B, and C, and we know the given relation between λ and μ:
<br/><br/>$$A = \lambda \hat{i} - \hat{j} + \hat{k}$$
<br/><br/>$$B = \hat{i} + 2 \hat{j} + ... |
<p>Let $$a, b, c$$ be three distinct real numbers, none equal to one. If the vectors $$a \hat{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \hat{j}+\hat{\mathrm{k}}$$ and $$\hat{\mathrm{i}}+\hat{\mathrm{j}}+c \hat{\mathrm{k}}$$ are coplanar, then $$\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$$ is equal to :</p... | ["B"]
Explanation:
$$
\left|\begin{array}{lll}
a & 1 & 1 \\\\
1 & \mathrm{~b} & 1 \\\\
1 & 1 & \mathrm{c}
\end{array}\right|=0
$$
<br/><br/>$$
\mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_1, \mathrm{C}_3 \rightarrow \mathrm{C}_3-\mathrm{C}_1
$$
<br/><br/>$$
\begin{aligned}
& \left|\begin{array}{lll}
a & 1-a & 1-a ... |
<p>Let $$\lambda \in \mathbb{Z}, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$$ and $$\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$$. Let $$\vec{c}$$ be a vector such that $$(\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=-17$$ and $$\vec{b} \cdot \vec{c}=-20$$. Then $$|\vec{c} \times(\lambda \hat{... | ["D"]
Explanation:
The given vectors are :
<br/><br/>$$\vec{a} = \lambda \hat{i} + \hat{j} - \hat{k}$$
<br/><br/>$$\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$$
<br/><br/>We are given that $(\vec{a} + \vec{b} + \vec{c}) \times \vec{c} = 0$ which implies $(\vec{a} + \vec{b}) \times \vec{c} = 0$. So, $\vec{c}$ is in the ... |
<p>If four distinct points with position vectors $$\vec{a}, \vec{b}, \vec{c}$$ and $$\vec{d}$$ are coplanar, then $$[\vec{a} \,\,\vec{b} \,\,\vec{c}]$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "$$[\\vec{d} \\,\\,\\,\\,\\,\\vec{b} \\,\\,\\,\\,\\,\\vec{a}]+[\\vec{a} \\,\\,\\,\\,\\,\\vec{c} \\,\\,\\,\\,... | ["D"]
Explanation:
$$
\begin{aligned}
& {[\vec{b}-\vec{a} \,\,\,\,\,\vec{c}-\vec{a} \,\,\,\,\,\vec{d}-\vec{a}]=0} \\\\
& (\vec{b}-\vec{a}) \cdot[(\vec{c}-\vec{a}) \times(\vec{d}-\vec{a})]=0 \\\\
& (\vec{b}-\vec{a}) \cdot(\vec{c} \times \vec{d}-\vec{c} \times \vec{a}-\vec{a} \times \vec{d})=0 \\\\
& {[\vec{b}\,\,\,\,\,... |
<p>Let the vectors $$\vec{u}_{1}=\hat{i}+\hat{j}+a \hat{k}, \vec{u}_{2}=\hat{i}+b \hat{j}+\hat{k}$$ and $$\vec{u}_{3}=c \hat{i}+\hat{j}+\hat{k}$$ be coplanar. If the vectors $$\vec{v}_{1}=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \vec{v}_{2}=a \hat{i}+(b+c) \hat{j}+a \hat{k}$$ and $$\vec{v}_{3}=b \hat{i}+b \hat{j}+(c+a) \hat{... | ["A"]
Explanation:
Since, $\vec{u}_1, \vec{u}_2, \vec{u}_3$ are coplanar.
<br/><br/>So, $\left[\begin{array}{lll}\vec{u}_1 & \vec{u}_2 & \vec{u}_3\end{array}\right]=0$
<br/><br/>$$
\begin{aligned}
& \Rightarrow\left|\begin{array}{lll}
1 & 1 & a \\
1 & b & 1 \\
c & 1 & 1
\end{array}\right|=0 \\\\
& \Rightarrow 1(b-1)-1... |
<p>Let the position vectors of the points A, B, C and D be
$$5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}$$ and $$-\hat{i}+5 \hat{j}+6 \hat{k}$$. Let the set $$S=\{\lambda \in \mathbb{R}$$ :
the points A, B, C and D are coplanar $$\}$$.
<br/><br/>Then $$\sum_\... | ["D"]
Explanation:
Given, position vectors of the points $A, B, C$ and $D$ be
<br/><br/>$5 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \lambda \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}},-2 \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+5 \hat{\mathbf{j}}+6... |
<p>Let the vectors $$\vec{a}, \vec{b}, \vec{c}$$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by $$\vec{a}, \vec{b}+\vec{c}$$ and $$\vec{a}+2 \vec{b}+3 \vec{c}$$ is equal to :</p>
Options:
[{"identifier": "A", "content... | ["D"]
Explanation:
Given that the volume $V$ of the parallelepiped formed by the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is represented by the scalar triple product $[\vec{a},\vec{b},\vec{c}]$, which is the determinant of the 3 x 3 matrix with vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ as its rows (or columns).
... |
<p>The sum of all values of $$\alpha$$, for which the points whose position vectors are $$\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{j}+4 \hat{k},(\alpha+1) \hat{i}+2 \hat{k}$$ and $$9 \hat{i}+(\alpha-8) \hat{j}+6 \hat{k}$$ are coplanar, is equal to :</p>
Options:
[{"identifier": "A", "content": "6"}, {"identifier"... | ["D"]
Explanation:
Let $\overrightarrow{O A}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
<br/><br/>$$
\begin{aligned}
& \overrightarrow{O B}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}} \\\\
& \overrightarrow{O C}=(a+1) \hat{\mathbf{i}}+2 \hat{\mathbf{k}}
\end{aligned}
$$
<br/><br/>and $ \overr... |
Let $\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+\hat{k}, $
<br/>$\overrightarrow{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k})$.
<br/>Let $\overrightarrow{\mathrm{c}}$ be the vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$ and $\vec{a} \cdot \vec{c}=3$.
<br/> T... | ["C"]
Explanation:
<p>$$\begin{aligned}
& \vec{a} \cdot[(\vec{c} \times \vec{b})-\vec{b}-\vec{c}] \\
& \vec{a} \cdot(\vec{c} \times \vec{b})-\vec{a} \cdot \vec{b}-\vec{a} \cdot \vec{c} \quad \text{..... (i)}
\end{aligned}$$</p>
<p>$$\begin{aligned}
& \text { given } \vec{a} \times \vec{c}=\vec{b} \\
& \Rightarrow(\vec... |
<p>Let $$\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$$ and $$\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}$$.
If $$\vec{d}$$ is the unit vector in the direction of $$\vec{b}+\vec{c}$$ such that $$\vec{a} \cdot \vec{d}=1$$, then $$(\vec{a} \times \vec{b}) \cdot \vec{c}$$ is equal to<... | ["C"]
Explanation:
<p>$$\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}, \vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in R$$<?p>
<p>$$\text { also, } \vec{b}+\vec{c}=(x+2) \hat{i}+6 \hat{j}-2 \hat{k}$$</p>
<p>$$\vec{d} \text { is the unit vector in the direction of } \vec{b}+\vec{c}$$</p>
<p>$$\beg... |
If $$\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3$$ thus what will be the value of $$\left| {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a } \right|,$$ given that $$\overrigh... | ["A"]
Explanation:
We have, $$\overrightarrow a + \overrightarrow 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 } ... |
$$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$ are $$3$$ vectors, such that <br/><br/>$$\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$$ , $$\left| {\overrightarrow a } \right| = 1\,\,\,\left| {\overrightarrow b } \right| = 2,\,\,\,\left| {\overrightarrow c } \right| = 3,$$,
<br/><b... | ["C"]
Explanation:
$$\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$$
<br><br>$$ \Rightarrow \left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right).\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) = 0$$
<br><br>$${\left| {\overrightarrow a } \right... |
A particle acted on by constant forces $$4\widehat i + \widehat j - 3\widehat k$$ and $$3\widehat i + \widehat j - \widehat k$$ is displaced from the point $$\widehat i + 2\widehat j + 3\widehat k$$ to the point $$\,5\widehat i + 4\widehat j + \widehat k.$$ The total work done by the forces is :
Options:
[{"identifie... | ["D"]
Explanation:
The work done by a force on a particle is given by the dot product of the force and the displacement vector of the particle. The displacement vector can be found by subtracting the initial position from the final position:
<br/><br/>
$$\mathbf{displacement} = \mathbf{final\ position} - \mathbf{initi... |
Let $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ be such that $$\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow v } \right|2,\,\,\,\left| {\overrightarrow w } \right|3.$$ If the projection $${\overrightarrow v }$$ along $${\overrightarrow u }$$ is equal to that of $${\overrightarro... | ["C"]
Explanation:
Projection of $$\overrightarrow v $$ along $$\overrightarrow u = {{\overrightarrow v .\overrightarrow u } \over {\left| {\overrightarrow u } \right|}} = {{\overrightarrow v .\overrightarrow u } \over 2}$$
<br><br>projection of $$\overrightarrow w $$ along $$\overrightarrow u = {{\overrightarrow w... |
The values of a, for which the points $$A, B, C$$ with position vectors $$2\widehat i - \widehat j + \widehat k,\,\,\widehat i - 3\widehat j - 5\widehat k$$ and $$a\widehat i - 3\widehat j + \widehat k$$ respectively are the vertices of a right angled triangle with $$C = {\pi \over 2}$$ are :
Options:
[{"identifier... | ["A"]
Explanation:
$$\overrightarrow {CA} = \left( {2 - a} \right)\widehat i + 2\widehat j;$$
<br><br>$$\overrightarrow {CB} = \left( {1 - a} \right)\widehat i - 6\widehat k$$
<br><br>$$\overrightarrow {CA} .\overrightarrow {CB} = 0$$
<br><br>$$\,\,\,\,\,\,\,\, \Rightarrow \left( {2 - a} \right)\left( {1 - a} \righ... |
If the vectors $$\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i + 4\widehat j + \widehat k\,\,\,$$ and $$\,\overrightarrow c = \lambda \widehat i + \widehat j + \mu \widehat k$$ are mutually orthogonal, then $$\,\left( {\lambda ,\mu } \right)$$ is equal to :
Opt... | ["D"]
Explanation:
Since, $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ are mutually orthogonal
<br><br> $$\overrightarrow a .\overrightarrow b = 0,\,\,\overrightarrow b .\overrightarrow c = 0,\,\,\overrightarrow c .\overrightarrow a = 0$$
<br><br>$$ \Rightarrow 2\lambda + 4 + \mu = 0\,\,\... |
Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two unit vectors. If the vectors $$\,\overrightarrow c = \widehat a + 2\widehat b$$ and $$\overrightarrow d = 5\widehat a - 4\widehat b$$ are perpendicular to each other, then the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is :
Options:
[{... | ["C"]
Explanation:
Let $$\overrightarrow c = \widehat a + 2\widehat b$$ and $$\overrightarrow d = 5\widehat a - 4\widehat b$$
<br><br>Since $$\overrightarrow c $$ and $$\overrightarrow d $$ are perpendicular to each other
<br><br>$$\therefore$$ $$\overrightarrow c .\overrightarrow d = 0 \Rightarrow \left( {\wideha... |
Let $$ABCD$$ be a parallelogram such that $$\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrightarrow p $$ and $$\angle BAD$$ be an acute angle. If $$\overrightarrow r $$ is the vector that coincide with the altitude directed from the vertex $$B$$ to the side $$AD,$$ then $$\overrightarrow r $... | ["B"]
Explanation:
Let $$ABCD$$ be a parallelogram such that
<br><br>$$\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrightarrow p $$ and $$\angle BAD$$ be an acute angle.
<br><br>We have
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263647/exa... |
In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively 3$$\widehat i$$ + $$\widehat j$$ $$-$$ $$\widehat k$$, $$-$$$$\widehat i$$ + 3$$\widehat j$$ + p$$\widehat k$$ and 5$$\widehat i$$ + q$$\widehat j$$ $$-$$ 4$$\widehat k$$, then the point (p, q) lies
on a line :
Op... | ["C"]
Explanation:
Given,
<br><br>$$\overrightarrow A = 3\widehat i + \widehat j - \widehat k$$
<br><br>$$\overrightarrow B = - \widehat i + 3\widehat j - p\widehat k$$
<br><br>$$\overrightarrow C = 5\widehat i + 9\widehat j - 4\widehat k$$
<br><br>$$ \therefore $$ $$\overrightarrow {AB} = - 4\... |
Let $$\overrightarrow u $$ be a vector coplanar with the vectors $$\overrightarrow a = 2\widehat i + 3\widehat j - \widehat k$$ and $$\overrightarrow b = \widehat j + \widehat k$$. If $$\overrightarrow u $$ is perpendicular to $$\overrightarrow a $$ and $$\overrightarrow u .\overrightarrow b = 24$$, then $${\left| {... | ["A"]
Explanation:
You should know that, when $$\overrightarrow u $$ is coplanar with $$\overrightarrow a $$ and $$\overrightarrow b $$ then we can write $$\overrightarrow u = x\overrightarrow a + y\overrightarrow b $$
<br><br>Here, $$\overrightarrow u $$ is perpendicular with $$\overrightarrow a $$ then,
<br><br... |
Let $$\sqrt 3 \widehat i + \widehat j,$$ $$\widehat i + \sqrt 3 \widehat j$$ and $$\beta \widehat i + \left( {1 - \beta } \right)\widehat j$$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is ... | ["B"]
Explanation:
Angle bisector is x $$-$$ y = 0
<br><br>$$ \Rightarrow $$ $${{\left| {\beta - \left( {1 - \beta } \right)} \right|} \over {\sqrt 2 }} = {3 \over {\sqrt 2 }}$$
<br><br>$$ \Rightarrow $$ $$\left| {2\beta - 1} \right| = 3$$
<br><br>$$ \Rightarrow $$ $$\beta $$ = 2 or ... |
Let A (3, 0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G
divides BM in the ratio, 2 : 1, then cos ($$\angle $$GOA) (O being the origin) is equal to :
Options:
[{"identifier": "A", "content": "$${1 \\over {\\sqrt {15} }}$$"}, {"identifier": "B", "content": "$${1 \\ove... | ["A"]
Explanation:
G is the centroid of $$\Delta $$ABC<br><br>
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266720/exam_images/loddnxhki4jx9azlbmpr.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v173426534... |
If a unit vector $$\overrightarrow a $$ makes angles $$\pi $$/3 with $$\widehat i$$ , $$\pi $$/ 4
with $$\widehat j$$ and $$\theta $$$$ \in $$(0, $$\pi $$) with $$\widehat k$$, then a value of $$\theta $$
is :-
Options:
[{"identifier": "A", "content": "$${{5\\pi } \\over {6}}$$"}, {"identifier": "B", "content": "$${{5... | ["C"]
Explanation:
A unit vector $$\overrightarrow a $$ makes angles $$\pi $$/3 with $$\widehat i$$
<br><br>$$ \therefore $$ $$\alpha $$ = $$\pi $$/3
<br><br> and $$\pi $$/ 4
with $$\widehat j$$
<br><br>$$ \therefore $$ $$\beta $$ = $$\pi $$/ 4
<br><br>and $$\theta $$$$ \in $$(0, $$\pi $$) with $$\widehat k$$
<br><br... |
Let $$\overrightarrow a = 2\widehat i + {\lambda _1}\widehat j + 3\widehat k,\,\,$$ $$\overrightarrow b = 4\widehat i + \left( {3 - {\lambda _2}} \right)\widehat j + 6\widehat k,$$ and $$\overrightarrow c = 3\widehat i + 6\widehat j + \left( {{\lambda _3} - 1} \right)\widehat k$$ be three vectors such that $$\o... | ["C"]
Explanation:
Given $$\overrightarrow b = 2\overrightarrow a $$
<br><br>$$ \therefore $$ $$4\widehat i + \left( {3 - {\lambda _2}} \right)\widehat j + 6\widehat k = 4\widehat i + 2{\lambda _1}\widehat j + 6\widehat k$$
<br><br>$$ \Rightarrow 3 - {\lambda _2} = 2{\lambda _1} \Rightarrow 2{\lambda _1} + {\lambda _... |
Let $$\overrightarrow a = \widehat i + \widehat j + \sqrt 2 \widehat k,$$ $$\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + \sqrt 2 \widehat k$$, $$\overrightarrow c = 5\widehat i + \widehat j + \sqrt 2 \widehat k$$ be three vectors such that the projection vector of $$\overrightarrow b $$ on $$\o... | ["B"]
Explanation:
Projection of $$\overrightarrow b $$ on $$\overrightarrow a $$ is $$\overrightarrow a $$
<br><br>$$ \therefore $$ $${{\overrightarrow b \cdot \overrightarrow a } \over {\left| {\overrightarrow a } \right|}} = \left| {\overrightarrow a } \right|$$
<br><br>$$ \Rightarrow $$ &nbs... |
A vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$$ lies in the plane of the vectors, $$\overrightarrow b = \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i - \widehat j + 4\widehat k$$. If $$\overrightarrow a $$ bisects the angle be... | ["B"]
Explanation:
Angle bisector $$\overrightarrow a = \lambda \left( {\widehat b + \widehat c} \right)$$
<br><br>= $$\lambda \left( {{{\widehat i + \widehat j} \over {\sqrt 2 }} + {{\widehat i - \widehat j + 4\widehat k} \over {3\sqrt 2 }}} \right)$$
<br><br>$$ \Rightarrow $$ $$\overrightarrow a = {\lambda \over ... |
Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that
<br/>$${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$$ = 8.
<br/><br/>Then $${\left| {\overrightarrow a + 2\overrightarrow b } ... | 2
Explanation:
Given, $$\left| {\overrightarrow a } \right| = \left| {\overrightarrow b } \right| = \left| {\overrightarrow c } \right| = 1$$
<br><br>$${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$$ = 8
<br><br>$$ \Rightarrow $$ $${\lef... |
Let a, b c $$ \in $$ R be such that a<sup>2</sup>
+ b<sup>2</sup>
+ c<sup>2</sup>
= 1. If <br/>$$a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}} \right) = c\cos \left( {\theta + {{4\pi } \over 3}} \right)$$,
<br/>where
$${\theta = {\pi \over 9}}$$, then the angle between the vectors
$$a\widehat i + b\w... | ["D"]
Explanation:
Let, $$\overrightarrow {{a_1}} = a\widehat i + b\widehat j + c\widehat k$$<br><br>and $$\overrightarrow {{a_2}} = b\widehat i + c\widehat j + a\widehat k$$<br><br>We know, Angle between two vectors<br><br>$$\cos \alpha = {{\overrightarrow {{a_1}} \,.\,\overrightarrow {{a_2}} } \over {|\overrighta... |
Let the vectors $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$
be such that
<br/>$$\left| {\overrightarrow a } \right| = 2$$, $$\left| {\overrightarrow b } \right| = 4$$
and $$\left| {\overrightarrow c } \right| = 4$$. If the projection of
<br/>$$\overrightarrow b $$
on $$\overrightarrow a $$... | 6
Explanation:
Projection of $$\overrightarrow b $$
on $$\overrightarrow a $$
= Projection of $$\overrightarrow c $$
on $$\overrightarrow a $$
<br><br>$$ \Rightarrow $$ $${{\overrightarrow b .\overrightarrow a } \over {\left| {\overrightarrow a } \right|}} = {{\overrightarrow c .\overrightarrow a } \over {\left| {\o... |
If $$\overrightarrow x $$ and $$\overrightarrow y $$ be two non-zero vectors such that
$$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$ and $${2\overrightarrow x + \lambda \overrightarrow y }$$ is perpendicular to $${\overrightarrow y }$$,
then the value of $$\lambda $... | 1
Explanation:
$$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$
<br>Squaring both sides we get
<br><br>$${\left| {\overrightarrow x } \right|^2} + 2\overrightarrow x .\overrightarrow y + {\left| {\overrightarrow y } \right|^2} = {\left| {\overrightarrow x } \right|^2}... |
Let $$\overrightarrow x $$ be a vector in the plane containing vectors $$\overrightarrow a = 2\widehat i - \widehat j + \widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$. If the vector $$\overrightarrow x $$ is perpendicular to $$\left( {3\widehat i + 2\widehat j - \widehat k} \right)$$ a... | 486
Explanation:
Let, $$\overrightarrow x = k(\overrightarrow a + \lambda \overrightarrow b )$$<br><br>$$\overrightarrow x$$ is perpendicular to $$3\widehat i + 2\widehat j - \widehat k$$<br><br><b>I.</b> k{(2 + $$\lambda$$)3 + (2$$\lambda$$ $$-$$ 1)2 + (1 $$-$$ $$\lambda$$)($$-$$1) = 0<br><br>$$ \Rightarrow $$ 8$$... |
In a triangle ABC, if $$|\overrightarrow {BC} | = 8,|\overrightarrow {CA} | = 7,|\overrightarrow {AB} | = 10$$, then the projection of the vector $$\overrightarrow {AB} $$ on $$\overrightarrow {AC} $$ is equal to :
Options:
[{"identifier": "A", "content": "$${{25} \\over 4}$$"}, {"identifier": "B", "content": "$${{127... | ["C"]
Explanation:
<picture><source media="(max-width: 1728px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266979/exam_images/tvnba426ygdio3l2ckwo.webp"><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264573/exam_images/vlfqrjsuvhpnf9m6eq5m.webp"><s... |
Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $$\theta$$, with the vector $$\overrightarrow a $$ + $$\overrightarrow b $$ + $$\overrightarrow c $$. Then 36cos<sup>2</sup>2$$\theta$$ is equal to __... | 4
Explanation:
$${\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2} = {\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow b } \right|^2} + {\left| {\overrightarrow c } \right|^2} + 2(\overrightarrow a \,.\,\overrightarrow b + \overrightarrow a \,.\,\overrightarrow c + \o... |
In a triangle ABC, if $$\left| {\overrightarrow {BC} } \right| = 3$$, $$\left| {\overrightarrow {CA} } \right| = 5$$ and $$\left| {\overrightarrow {BA} } \right| = 7$$, then the projection of the vector $$\overrightarrow {BA} $$ on $$\overrightarrow {BC} $$ is equal to :
Options:
[{"identifier": "A", "content": "$${{1... | ["C"]
Explanation:
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264070/exam_images/ep6a24u2iigywhjyduzo.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 20th July Evening Shift Mathematics - Vector Algebra Question 131 English Expl... |
For p > 0, a vector $${\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j$$ is obtained by rotating the vector $${\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j$$ by an angle $$\theta$$ about origin in counter clockwise direction. If $$\tan \theta = {{\left( {\alpha \sqrt 3 - 2} \right)} \over {\lef... | 6
Explanation:
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265929/exam_images/pfsd2oapdy3p6fqplgh1.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264150/exam_images/yosxjcnh7ins9tydvyvk.webp"><img sr... |
If $$\left( {\overrightarrow a + 3\overrightarrow b } \right)$$ is perpendicular to $$\left( {7\overrightarrow a - 5\overrightarrow b } \right)$$ and $$\left( {\overrightarrow a - 4\overrightarrow b } \right)$$ is perpendicular to $$\left( {7\overrightarrow a - 2\overrightarrow b } \right)$$, then the angle between... | 60
Explanation:
$$\left( {\overrightarrow a + 3\overrightarrow b } \right) \bot \left( {7\overrightarrow a - 5\overrightarrow b } \right)$$<br><br>$$ \therefore $$ $$\left( {\overrightarrow a + 3\overrightarrow b } \right)\,.\,\left( {7\overrightarrow a - 5\overrightarrow b } \right) = 0$$<br><br>$$ \Rightarrow $$... |
A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $${\cos ^{ - 1}}{1 \over 5}$$, then the height of the hall (in meters) is :<br/><br/><img src="data:image/png;base64,UklGRkQOAABXRUJQVlA4IDgOAADwVwCdASocARIBPm00l0ekIyKhJX... | ["D"]
Explanation:
$$A(\widehat j)\,.\,B(10\widehat i)$$<br><br>$$H(h\widehat j + 10\widehat k)$$<br><br>$$G(10\widehat i + h\widehat j + 10\widehat k)$$<br><br>$$\overrightarrow {AG} = 10\widehat i + h\widehat j + 10\widehat k$$<br><br>$$\overrightarrow {BH} = - 10\widehat i + h\widehat j + 10\widehat k$$<br><br>$... |
If the projection of the vector $$\widehat i + 2\widehat j + \widehat k$$ on the sum of the two vectors $$2\widehat i + 4\widehat j - 5\widehat k$$ and $$ - \lambda \widehat i + 2\widehat j + 3\widehat k$$ is 1, then $$\lambda$$ is equal to __________.
Options:
[] | 5
Explanation:
$$\overrightarrow a = \widehat i + 2\widehat j + \widehat k$$<br><br>$$\overrightarrow b = (2 - \lambda )\widehat i + 6\widehat j - 2\widehat k$$<br><br>$${{\overrightarrow a \,.\,\overrightarrow b } \over {|\overrightarrow b |}} = 1,\overrightarrow a \,.\,\overrightarrow b = 12 - \lambda $$<br><br>$... |
Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two vectors <br/>such that $$\left| {2\overrightarrow a + 3\overrightarrow b } \right| = \left| {3\overrightarrow a + \overrightarrow b } \right|$$ and the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is 60$$^\circ$$. If $${1 \over 8}\overrig... | ["C"]
Explanation:
$${\left| {3\overrightarrow a + \overrightarrow b } \right|^2} = {\left| {2\overrightarrow a + 3\overrightarrow b } \right|^2}$$<br><br>$$\left( {3\overrightarrow a + \overrightarrow b } \right).\left( {3\overrightarrow a + \overrightarrow b } \right) = \left( {2\overrightarrow a + 3\overrighta... |
<p>Let $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$$ $${a_i} > 0$$, $$i = 1,2,3$$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $$\overrightarrow a $$ on the vector $$3\widehat i + 4\widehat j$$ be 7. Let $$\overrightarrow b $$... | ["B"]
Explanation:
<p>$${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1 \Rightarrow {\cos ^2}\alpha = {1 \over 3} \Rightarrow \cos \alpha = {1 \over {\sqrt 3 }}$$</p>
<p>$$\overrightarrow a = {\lambda \over 3}(\widehat i + \widehat j + \widehat k),\,\lambda > 0$$</p>
<p>$${\lambda \over {\sqrt 3 }}{{(\... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.