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: 6

Answer

${\bf u}\times{\bf v}$ has length $1$ and direction ${\bf j}$ ${\bf v}\times{\bf u}$ has length $1$ and direction $-{\bf j}$

Work Step by Step

${\bf u}={\bf i}\times{\bf j}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{array}\right|={\bf k}$ ${\bf v}={\bf j}\times{\bf k}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right|={\bf i}$ ${\bf w}={\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{array}\right|=0-(0-1){\bf j}+0={\bf j}$ $\left|{\bf w}\right|=1,$ ${\bf u}\times{\bf v}$ has length $1$ and direction ${\bf j}$ By property 3 (see "Properties of the Cross Product" box on p. 618) ${\bf v}\times{\bf u}=-{\bf w}=1\cdot(-{\bf j})$ ${\bf v}\times{\bf u}$ has length $1$ and direction $-{\bf j}$
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