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Unformatted text preview: point as u. Completing the parallelogram, we see that
u v v u. This also gives another way to construct the sum: If we place u and v so
they start at the same point, then u v lies along the diagonal of the parallelogram with
u and v as sides. (This is called the Parallelogram Law.)
EXAMPLE 1 Draw the sum of the vectors a and b shown in Figure 5.
a b SOLUTION First we translate b and place its tail at the tip of a, being careful to draw a copy
of b that has the same length and direction. Then we draw the vector a b [see 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 b by the
Parallelogram Law as in Figure 6(b). FIGURE 5 a
Visual 13.2 shows how the Triangle and
Parallelogram Laws work for various
vectors u and v. FIGURE 6 a b
a+b a+b (a) b (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 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 mul- 2v v _v _1.5 v FIGURE 7 Scalar multiples of v 1
2v tiple 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
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 v
1 v has the same
length as v but points in the opposite direction. We call it the negative of v.
By the difference u v of two vectors we mean
u v u v 5E-13(pp 828-837) 836 ❙❙❙❙ 1/18/06 11:11 AM Page 836 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE So we can construct u v by ﬁrst drawing the negative of v, v, and then adding it to u
by the Parallelogram Law as in Fi...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
- Fall '09