Finding a Vector Orthogonal to Two Given Vectors

Find a vector orthogonal to the vectors \(\mathbf{v}=2\mathbf{i}-\mathbf{j}+ \mathbf{k}\) and \(\mathbf{w}=2\mathbf{i}+2\mathbf{j}-\mathbf{k}\).

Solution Since \(\mathbf{v\times w}\) is orthogonal to both the vector \(\mathbf{v}\) and the vector \(\mathbf{w}\), a vector orthogonal to \( \mathbf{v}\) and \(\mathbf{w}\) is \[ \mathbf{v}\times \mathbf{w}= \left|\begin{array}{c@{\quad}c@{\quad}c} \mathbf{i} & \hphantom{-}\mathbf{j} & \hphantom{-}\mathbf{k} \\[3pt] 2 & -1 & \hphantom{-}1 \\[3pt] 2 & \hphantom{-}2 & -1 \end{array}\right| =-\mathbf{i}+4\mathbf{j}+6\mathbf{k} \]