James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 12.2 Vectors 799
Figure 6(a)] starting at the initial point of a and ending at the terminal point of the copy
of b.
Alternatively, we could place b so it starts where a starts and construct a 1 b by the
Parallelogram Law as in Figure 6(b).
TEC Visual 12.2 shows how the Triangle
and Parallelogram Laws work
for various vectors a and b.
b
a
a+b
a
a+b
b
FIGURE 6
(a)
(b) ■
It is possible to multiply a vector by a real number c. (In this context we call the real number
c a scalar to distinguish it from a vector.) For instance, we want 2v to be the same
vector as v 1 v, which has the same direction as v but is twice as long. In general, we multiply
a vector by a scalar as follows.
Definition of Scalar Multiplication If c is a scalar and v is a vector, then the
scalar multiple cv is the vector whose length is | c | times the length of v and
whose direction is the same as v if c . 0 and is opposite to v if c , 0. If c − 0 or
v − 0, then cv − 0.
This definition is illustrated in Figure 7. We see that real numbers work like scaling factors
here; that’s why we call them scalars. Notice that two nonzero vectors are parallel
if they are scalar multiples of one another. In particular, the vector 2v − s21dv has the
same length as v but points in the opposite direction. We call it the negative of v.
FIGURE 7
Scalar multiples of v
v 2v 1
2 v _v
_1.5v
By the difference u 2 v of two vectors we mean
u 2 v − u 1 s2vd
So we can construct u 2 v by first drawing the negative of v, 2v, and then adding it to
u by the Parallelogram Law as in Figure 8(a). Alternatively, since v 1 su 2 vd − u,
the vector u 2 v, when added to v, gives u. So we could construct u 2 v as in Fig ure
8(b) by means of the Triangle Law. Notice that if u and v both start from the same initial
point, then u 2 v connects the tip of v to the tip of u.
v
u
u-v
u-v
FIGURE 8
Drawing u 2 v
_v
(a)
v
(b)
u
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