Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 12: Vectors and the Geometry of Space - Section 12.4 - The Cross Product - Exercises 12.4 - Page 719: 16


$a.\quad 3$ $b.\displaystyle \quad \frac{2}{3} {\bf i}+ \frac{2}{3} {\bf j}- \frac{1}{3} {\bf k}$

Work Step by Step

${\bf u}= \overrightarrow{PQ} = \langle 2-1,1-1,3-1 \rangle= \langle 1,0,2 \rangle$ ${\bf v}= \overrightarrow{PR} = \langle 3-1,-1-1,1-1 \rangle= \langle 2,-2,0 \rangle$ We find the area of the parallelogram $|{\bf u}\times{\bf v}|$ The area of the triangle is half the area of the parallelogram. $A=\displaystyle \frac{1}{2}\cdot|{\bf u}\times{\bf v}|$ ${\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 1 & 0 & 2\\ 2 & -2 & 0 \end{array}\right|=(0+4){\bf i}-(0-4{\bf j}+(-2-0){\bf k}$ $=4{\bf i}+4{\bf j}-2{\bf k}$ $|{\bf u}\times{\bf v}|=\sqrt{16+16+4}=\sqrt{36}=6$ $A=\displaystyle \frac{1}{2}\cdot 6=3$ $(b)$ ${\bf w}={\bf u}\times{\bf v}$ is perpendicular to both ${\bf u}$ and ${\bf v}$ (and the plane they belong to). A unit vector has length 1, so we take $\displaystyle \frac{{\bf w}}{|{\bf w}|}= \frac{1}{6} (4{\bf i}+4{\bf j}-2{\bf k})$ $= \displaystyle \frac{2}{3} {\bf i}+ \frac{2}{3} {\bf j}- \frac{1}{3} {\bf k}$
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