Precalculus (6th Edition) Blitzer

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

Chapter 6 - Section 6.6 - Vectors - Exercise Set - Page 782: 57



Work Step by Step

Vectors are $u={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j},v={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j},$ and $w={{a}_{3}}\mathbf{i}+{{b}_{3}}\mathbf{j}$. Now, put these vectors in the expression $u+v=v+u$. Then, $\begin{align} & \left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)+\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right)=\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right)+\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right) \\ & \left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}=\left( {{a}_{2}}+{{a}_{1}} \right)\mathbf{i}+\left( {{b}_{2}}+{{b}_{1}} \right)\mathbf{j} \\ & \left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \\ \end{align}$ Left side and right side are equal. Hence, $u+v=v+u$.
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