Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 794: 70


Two vectors are orthogonal if the dot product of both the vectors is 0.

Work Step by Step

If $v$ and $w$ are non-zero vectors and $\theta $ is the smallest non-negative angle between $\mathbf{u}$ and $\mathbf{w}$, Then the angle between two vectors can be calculated as: $\begin{align} & \cos \theta =\frac{\mathbf{u}\centerdot \mathbf{w}}{\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|} \\ & \mathbf{u}\centerdot \mathbf{w}=\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|\cos \theta \\ \end{align}$ Two vectors are said to be orthogonal if the angle between them is ${{90}^{\circ }}$ If $\mathbf{u}$ and $\mathbf{w}$ are orthogonal then, $\begin{align} & \mathbf{u}\centerdot \mathbf{w}=\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|\cos 90{}^\circ \\ & =\left\| \mathbf{u} \right\|\left\| \mathbf{w} \right\|\left( 0 \right) \\ & =0 \end{align}$ $\mathbf{u}\centerdot \mathbf{w}=0$
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