## Precalculus (6th Edition) Blitzer

Dot proct of vectors: For two vectors $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$ and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$ $\mathbf{v}\cdot \mathbf{w}={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}$. If the dot product of two nonzero vectors is zero then the vectors are said to be orthogonal vectors. Here, ${{a}_{1}}=12,{{a}_{2}}=2,{{b}_{1}}=-8,{{b}_{2}}=3$. So, \begin{align} & \mathbf{v}\cdot \mathbf{w}={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}} \\ & =12\left( 2 \right)+\left( -8 \right)\left( 3 \right) \\ & =24-24 \\ & =0 \end{align} Since, the dot product is zero so the vectors are orthogonal vectors.