Show that the vectors $\mathbf{v}=2\mathbf{i}+\mathbf{j}-\mathbf{k}$ and $\mathbf{w}=-4\mathbf{i}-2\mathbf{j}+2\mathbf{k}$ are parallel.

Solution We show that $\mathbf{v}\times \mathbf{w}=\mathbf{0}$. $$\begin{eqnarray*} \mathbf{v}\times \mathbf{w}&=&\left\vert \begin{array}{r@{\quad}r@{\quad}r} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 1 & -1 \\ -4 & -2 & 2 \end{array} \right\vert =\left\vert \begin{array}{r@{\quad}r} 1 & -1 \\ -2 & 2 \end{array} \right\vert \,\mathbf{i}-\left\vert \begin{array}{r@{\quad}r} 2 & -1 \\ -4 & 2 \end{array} \right\vert \,\mathbf{j}+\left\vert \begin{array}{r@{\quad}r} 2 & 1 \\ -4 & -2 \end{array} \right\vert \,\mathbf{k}\\[5pt] &=&0\mathbf{i}+0\mathbf{j}+0\mathbf{k}=\mathbf{0} \end{eqnarray*}$$