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 - Questions to Guide Your Review - Page 733: 4


See the explanation below.

Work Step by Step

${\bf u}=\langle u_{1}, u_{2}, u_{3}\rangle$ has magnitude $|{\bf u}|=\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}$ and direction $\displaystyle \frac{1}{\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}}\cdot\langle u_{1}, u_{2}, u_{3}\rangle$ Multiplying by a positive scalar, $ c{\bf u}=c\langle u_{1}, u_{2}, u_{3}\rangle=\langle cu_{1}, cu_{2}, cu_{3}\rangle$ $c{\bf u} $ has the same direction as ${\bf u}$ and its magnitude changes by factor $c$. If the scalar $c$ is zero, the vector $c{\bf u}$ is the zero vector, and has no direction, with magnitude of 0. If the scalar is negative, the direction of $c{\bf u}$ is opposite to the direction of ${\bf u}$ and the magnitude changes by factor $|c|.$
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