# Chapter 13

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 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 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...
View Full Document

## 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.

Ask a homework question - tutors are online