Find a scalar $a$ so that the vectors $\mathbf{v}=2a\mathbf{i}+\mathbf{j}- \mathbf{k}$ and $\mathbf{w}=\mathbf{i}-a \mathbf{j}+\mathbf{k}$ are orthogonal.

Solution The vectors $\mathbf{v}$ and $\mathbf{w}$ are orthogonal if $\mathbf{v}\,{\cdot}\, \mathbf{w}=0.$ So, $$\begin{eqnarray*} \mathbf{v}\,{\cdot}\, \mathbf{w}=2a-a-1&=& 0 \\[3pt] a&=& 1 \end{eqnarray*}$$

The vectors $\mathbf{v}=2\mathbf{i}+\mathbf{j}-\mathbf{k}$ and $\mathbf{w}= \mathbf{i}-\mathbf{j}+\mathbf{k}$ are orthogonal.