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which in some particular coordinate system has the three components (đ„, đ„, đ;),
and that b is another vector which has the three components (b„, b„,b„). Ñow
let us invent three new numbers (a„ + Ö„,d„ + b„,a„ + b;). Do these form a
vector? “Well,” we might say, “they are three numbers, and every three numbers
form a vector.” Ño, no£ every three numbers form a vector! In order for it to be
a vector, not only must there be three numbers, but these must be associated
with a coordinate system in such a way that 1Ý we turn the coordinate system,
the three numbers “revolve” on each other, get “mixed up” in each other, by
the precise laws we have already described. So the question is, if we now rotate
the coordinate system so that (a„, #,,ø„) become (đx, đ„,øz:) and (b„, b„, b„)
--- Trang 222 ---
become (b„¿, b„/, b„;), what do (az„ + bạ, ay + bự, ø„ + b„) become? Do they become
(Ga + bạ, dự -E bu,a„; + by) or not? The answer is, oÝ course, yes, because
the prototype transformations of Eq. (11.5) constitute what we call a ii¿mear
transformation. If we apply those transformations to ø„ and Ö„ to get aœ„ + bạ,
we fnd that the transformed a„ -+ b„ is indeed the same as ø„; + b„;. When œ
and ö are “added together” in this sense, they will form a vector which we may
call e. We would write this as
c=œa+b.
Now e has the interesting property
c=b+ea,
as we can immediately see from i%s components. 'Thus also,
œ-+(b+c)=(a+b)+c.
W© can add vectors in any order.
What is the geometric significance of œ + b? Suppose that œ and b were
represented by lines on a piece of paper, what would e look like? 'This is shown
in Eig. II-4. We see that we can add the components of b to those oŸ œ most
conveniently if we place the rectangle representing the components of next to
that representing the components of ø in the manner indicated. Since b just
“fñts” into its rectangle, as does ø into its rectangle, this is the same as putting
the “tail” of b on the “head” of ø, the arrow from the “tail” of œ to the “head”
of b being the vector œ. OÝ course, if we added ø to b the other way around, we
___— '
__— /. __—_— Ị
I 1 Xx
Fig. 11-4. The addition of vectors.
--- Trang 223 ---
would put the “tail” of œ on the “head” of b, and by the geometrical properties
of parallelograms we would get the same result for c. Note that vectors can be
added in this way without reference to any coordinate axes.
Suppose we multiply a vector by a number œ, what does this mean? We
deffne it to mean a new vector whose components are œø„, œa„, and œaz;. We
leave 1t as a problem for the student to prove that it 7s a vector.
Now let us consider vector subtraction. We may deñne subtraction in the
same way as addition, but instead of adding, we subtract the components. Ôr
we might defñne subtraction by defning a negative vector, —b = —1b, and then
we would add the components. ÏIt comes to the same thing. The result ¡is shown
in Eig. 11-5. This fñgure shows d = œ— b= ø-+ (—Ùb); we also note that the
diference ø — b can be found very easily from ø and b by using the equivalent
relatlon œ = b+ d. 'Thus the diference is even easier to find than the sum: we
Jjust draw the vector from b to ø, to get œ — bÌ
Fig. 11-5. The subtraction of vectors.
Next we discuss velocity. Why is velocity a vector? lÝ position is given
by the three coordinates (z,,z), what is the velocity? "The velocity is given
by dz/dt, dụ/dt, and dz/dt. Is that a vector, or not? We can fnd out by
differentiating the expressions in Eq. (11.5) to find out whether đ+ /đ transƒorms
in the ripht way. We see that the components đz/đt and dụ/dt do transform
according to the same law as # and , and therefore the time derivative 2s a
vector. 5o the velocity is a vector. We can write the velocity in an interesting
WayV aS
= dr(dt.
What the velocity is, and why i% is a vector, can also be understood more
pictorially: How far does a particle move in a short time A£? Answer: Az, so if
a particle is “here” at one instant and “there” at another instant, then the vector
diference of the positions Am = rs — r, which is in the direction oŸ motion
--- Trang 224 ---
shown in Fig. 11-6, divided by the time interval Af = ‡a — f, is the “average
velocity” vector.
Ar = ra —
T2 1
Fig. 11-6. The displacement of a particle in a short time interval Af =
ta — tì.
In other words, by vector velocity we mean the limit, as A# goes to 0, of the
diference between the radius vectors at the time £ + A£ and the time ý, divided
by Ai:
ò= lim (Ar/At) = dr/át. (11.10)
Thus velocity is a vector because it is the difference of two vectors. ÏIt is also the
right defnition of velocity because its components are d+/dt, dụ/dt, and dz/di.
In fact, we see from this argument that if we diferentiate amw vector with respect
to time we produce a new vector. So we have several ways of producing new
vectors: (1) multiply by a constant, (2) diferentiate with respect to time, (3) add
or subtract bwo vectOrs.
11-6 Newton°s laws in vector notation
In order to write Newton”s laws in vector form, we have to go Just one step
further, and defne the acceleration vector. 'This is the time derivative of the
velocity vector, and it is easy to demonstrate that its components are the second
derivatives of z, , and z with respect to ý:
đu đÀ (dr đ?r
dt dt dt đị2
duy d2z đuy dầu dù; d2z
TC dị) C9 CA d) “5” đÐ — dự 112)
--- Trang 225 ---
With this defñnition, then, Newton's laws can be written in this way:
ma = F (11.13)
m(dŠr/dt?) = F. (11.14)
Now the problem of proving the invariance of Ñewton”s laws under rotation
of coordinates is this: prove that œ is a vector; this we have just done. Prove
that #' is a vector; we swppose it is. So 1Í Íforce is a vector, then, since we know
acceleration is a vector, q. (11.13) will look the same in any coordinate system.
Writing ¡it in a form which does not explicitly contain zø”s, 's, and zˆs has the