Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 12 - Section 12.3 - The Dot Product - 12.3 Exercises: 42

Answer

$\frac{8}{9}$, $\lt\frac{-8}{81}, \frac{32}{81},\frac{64}{81}\gt$

Work Step by Step

Given: $a=\lt-1,4,8\gt$ , $b=\lt12,1,2\gt$ Scalar Projection $b$ onto $a$ can be calculated as follows: $\frac{a \times b }{|a|}=\frac{(-1 \times 12)+( 4 \times 1)+(8 \times 2)}{\sqrt {{(-1)^{2}+(4)^{2}}+(8)^{2}}}$ $=\frac{-12+4+16}{\sqrt {81}}$ $=\frac{8}{9}$ Vector Projection $b$ onto $a$ can be calculated as follows: $\frac{a \times b }{|a|^{2}}\times a=\frac{8}{81}\lt-1,4,8\gt$ $=\lt\frac{-8}{81}, \frac{32}{81},\frac{64}{81}\gt$ Hence, Scalar Projection $b$ onto $a$ = $\frac{8}{9}$, Vector Projection $b$ onto $a$=$\lt\frac{-8}{81}, \frac{32}{81},\frac{64}{81}\gt$
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