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 - Concept and Vocabulary Check - Page 792: 1


If $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$ and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$ are vectors, the product $\mathbf{v}\cdot \mathbf{w}$, called the dot product, is defined as $\mathbf{v}\cdot \mathbf{w}={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}$.

Work Step by Step

Dot product of two vectors can be obtained as $\begin{align} & \mathbf{v}\cdot \mathbf{w}=\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)\cdot \left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right) \\ & ={{a}_{1}}{{a}_{2}}\left( \mathbf{i}\cdot \mathbf{i} \right)+{{a}_{1}}{{b}_{2}}\left( \mathbf{i}\cdot \mathbf{j} \right)+{{b}_{1}}{{a}_{2}}\left( \mathbf{j}\cdot \mathbf{i} \right)+{{b}_{1}}{{b}_{2}}\left( \mathbf{j}\cdot \mathbf{j} \right) \\ & ={{a}_{1}}{{a}_{2}}\left( 1 \right)+{{a}_{1}}{{b}_{2}}\left( 0 \right)+{{b}_{1}}{{a}_{2}}\left( 0 \right)+{{b}_{1}}{{b}_{2}}\left( 1 \right) \\ & ={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}} \end{align}$
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