Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 5 - Inner Product Spaces - Review Exercises - Page 284: 13

Answer

(a) $$u =(4,4,3).$$ (b)$$u =(-4,-4,-3).$$ (c) $$u =(-16,-16,-12).$$

Work Step by Step

Let $ v=(8,8,6)$, then we have (a) Since $u$ in the direction of $v$ then $u=cv$ ($c>0$). Since the length of $u$ is one-half of $v$, then $c=\frac{1}{2}$. Hence, $$u=\frac{1}{2}(8,8,6)=(4,4,3).$$ (b) Since $u$ in the direction opposite that of $v$ then $u=-cv$ ($c>0$). Since the length of $u$ is one-fourth of $v$, then $c=\frac{1}{4}$. Hence, $$u=-\frac{1}{4}(8,8,6)=(-4,-4,-3).$$ (c) Since $u$ in the direction opposite that of $v$ then $u=-cv$ ($c>0$). Since the length of $u$ is twice of $v$, then $c=2$. Hence, $$u=-2(8,8,6)=(-16,-16,-12).$$
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