University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 11 - Section 11.4 - The Cross Product - Exercises - Page 622: 1

Answer

${\bf u}\times{\bf v}$ has length $3$ and direction $\displaystyle \frac{2}{3}{\bf i}+\frac{1}{3}{\bf j}+\frac{2}{3}{\bf k}$ ${\bf v}\times{\bf u}$ has length $3$ and direction $-\displaystyle \frac{2}{3}{\bf i}-\frac{1}{3}{\bf j}-\frac{2}{3}{\bf k}$

Work Step by Step

${\bf w}={\bf u}\times{\bf v}=\left|\begin{array}{rrr} {\bf i} & {\bf j} & {\bf k}\\ 2 & -2 & -1\\ 1 & 0 & -1 \end{array}\right|$ $=(2-0){\bf i}-(-2+1){\bf j}+(0+2){\bf k}$ $=2{\bf i}+{\bf j}+2{\bf k}$ $|{\bf w}|=\sqrt{4+1+4}=3$ and the unit vector parallel to ${\bf w}$ is $\displaystyle \frac{{\bf w} }{|{\bf w} |}=\frac{2}{3}{\bf i}+\frac{1}{3}{\bf j}+\frac{2}{3}{\bf k}$ ${\bf w}=3(\displaystyle \frac{2}{3}{\bf i}+\frac{1}{3}{\bf j}+\frac{2}{3}{\bf k})$ ${\bf u}\times{\bf v}$ has length $3$ and direction $\displaystyle \frac{2}{3}{\bf i}+\frac{1}{3}{\bf j}+\frac{2}{3}{\bf k}$ By property 3 (see "Properties of the Cross Product" box on p. 618) ${\bf v}\displaystyle \times{\bf u}=-{\bf w}=3(-\frac{2}{3}{\bf i}-\frac{1}{3}{\bf j}-\frac{2}{3}{\bf k})$ ${\bf v}\times{\bf u}$ has length $3$ and direction $-\displaystyle \frac{2}{3}{\bf i}-\frac{1}{3}{\bf j}-\frac{2}{3}{\bf k}$
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