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 $$\mathbf{v}\times \mathbf{w}= \left|\begin{array}{c@{\quad}c@{\quad}c} \mathbf{i} & \hphantom{-}\mathbf{j} & \hphantom{-}\mathbf{k} \\[3pt] 2 & -1 & \hphantom{-}1 \\[3pt] 2 & \hphantom{-}2 & -1 \end{array}\right| =-\mathbf{i}+4\mathbf{j}+6\mathbf{k}$$