Making Two Vectors Orthogonal

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.