Chapter 13

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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 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...
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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.

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