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: 59


See the explanation below.

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 $c\left( u+v \right)=cu+cv$. Then, we get $\begin{align} & c\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}+{{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right)=c\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)+c\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right) \\ & c\left[ \left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \right]=c{{a}_{1}}\mathbf{i}+c{{b}_{1}}\mathbf{j}+c{{a}_{2}}\mathbf{i}+c{{b}_{2}}\mathbf{j} \\ & c{{a}_{1}}\mathbf{i}+c{{a}_{2}}\mathbf{i}+c{{b}_{1}}\mathbf{i}+c{{b}_{2}}\mathbf{j}=c{{a}_{1}}\mathbf{i}+c{{a}_{2}}\mathbf{i}+c{{b}_{1}}\mathbf{j}+c{{b}_{2}}\mathbf{j} \end{align}$ Left side and right side are equal. Hence, $c\left( u+v \right)=cu+cv$.
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