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 - 5.1 Length and Dot Product in Rn - 5.1 Exercises - Page 235: 31

Answer

(a) $\|u\|=\sqrt{3}, \quad \|v\|=2$. (b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\left( -\frac{1}{2}, \frac{\sqrt{2}}{2},\frac{1}{2} \right)$. (c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{\sqrt{3}}{3}\left(0, 1, \sqrt{2}\right)$. (d) ${u} \cdot {v} =0$. (e)${u} \cdot {u}=3$. (f) ${v} \cdot {v}=4$.

Work Step by Step

(a) $\|u\|=\sqrt{3}, \quad \|v\|=2$. (b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\left( -\frac{1}{2}, \frac{\sqrt{2}}{2},\frac{1}{2} \right)$. (c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{\sqrt{3}}{3}\left(0, 1, \sqrt{2}\right)$. (d) ${u} \cdot {v} =0$. (e)${u} \cdot {u}=3$. (f) ${v} \cdot {v}=4$.
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