Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 12 - Vectors and the Geometry of Space - 12.3 The Dot Product - 12.3 Exercises: 41

Answer

$\frac{1}{9}$, $\lt\frac{4}{81}, \frac{7}{81},\frac{-4}{81}\gt$

Work Step by Step

Given: $a=\lt4,7,-4\gt$ , $b=\lt3,-1,1\gt$ Scalar Projection $b$ onto $a$ can be calculated as follows: $\frac{a \times b }{|a|}=\frac{(4 \times 3)+( 7 \times -1)+(-4 \times 1)}{\sqrt {{(4)^{2}+(7)^{2}}+(-4)^{2}}}$ $=\frac{12-7-4}{\sqrt {81}}$ $=\frac{1}{9}$ Vector Projection $b$ onto $a$ can be calculated as follows: $\frac{a \times b }{|a|^{2}}\times a=\frac{1}{81}\lt4,7-4\gt$ $=\lt\frac{4}{81}, \frac{7}{81},\frac{-4}{81}\gt$ Hence, Scalar Projection $b$ onto $a$ = $\frac{1}{9}$, Vector Projection $b$ onto $a$=$\lt\frac{4}{81}, \frac{7}{81},\frac{-4}{81}\gt$
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