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 799: 91



Work Step by Step

Unit vector: For any vector $\mathbf{v}$, $\frac{\mathbf{v}}{\left\| \mathbf{v} \right\|}$ is the unit vector in the same direction as the vector $\mathbf{v}$. Magnitude of vector is given by: The magnitude of $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ is given by $\left\| \mathbf{v} \right\|=\sqrt{{{a}^{2}}+{{b}^{2}}}$. Scalar multiplication of a vector is given by: For any vector $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ and $k$ is a real number, $k\mathbf{v}=\left( ka \right)\mathbf{i}+\left( kb \right)\mathbf{j}$. Here, $\mathbf{v}=8\mathbf{i}-6\mathbf{j}$. So, $\begin{align} & \left\| \mathbf{v} \right\|=\sqrt{{{8}^{2}}+{{\left( -6 \right)}^{2}}} \\ & =\sqrt{64+36} \\ & =\sqrt{100} \\ & =10 \end{align}$ The unit vector in the direction as the vector $\mathbf{v}$ is calculated as below: $\begin{align} & \frac{\mathbf{v}}{\left\| \mathbf{v} \right\|}=\frac{8\mathbf{i}-6\mathbf{j}}{10} \\ & =\frac{8}{10}\mathbf{i}-\frac{6}{10}\mathbf{j} \\ & =\frac{4}{5}\mathbf{i}-\frac{3}{5}\mathbf{j} \end{align}$ Hence, the unit vector in the direction as the vector $\mathbf{v}$ is $\frac{4}{5}\mathbf{i}-\frac{3}{5}\mathbf{j}$.
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