## Precalculus (6th Edition) Blitzer

If $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$, and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$, then $\mathbf{v}+\mathbf{w}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}$ $\mathbf{v}-\mathbf{w}=\left( {{a}_{1}}-{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}-{{b}_{2}} \right)\mathbf{j}$ $k\mathbf{v}=k{{a}_{1}}\mathbf{i}+k{{a}_{2}}\mathbf{j}$
The values of the above expression can be calculated by substituting the values of $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$, and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$ as below: \begin{align} & \mathbf{v}+\mathbf{w}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}+{{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \\ & =\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \end{align} \begin{align} & \mathbf{v}-\mathbf{w}=\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)-\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right) \\ & =\left( {{a}_{1}}-{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}-{{b}_{2}} \right)\mathbf{j} \end{align} And, \begin{align} & k\mathbf{v}=k\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right) \\ & =k{{a}_{1}}\mathbf{i}+k{{a}_{2}}\mathbf{j} \end{align}