Showing Two Vectors Are Parallel

1. The vectors \[ \begin{equation*} \mathbf{v}=\langle 2,3,-1\rangle\qquad\hbox{and}\qquad\mathbf{w}=\langle 4,6,-2\rangle \end{equation*} \] are parallel, since \(\mathbf{v}=\dfrac{1}{2}\mathbf{w}\). In this case, \(\mathbf{v}\) and \(\mathbf{w}\) have the same direction.
2. The vectors \[ \begin{equation*} \mathbf{v}=\left\langle 1,2,3\right\rangle\qquad \hbox{and}\qquad\mathbf{w}=\langle -3,-6,-9\rangle \end{equation*} \] are parallel, since \(\mathbf{w}=-3\,\mathbf{v}\). In this case, \(\mathbf{v}\) and \(\mathbf{w}\) have opposite directions.