The vectors \(\mathbf{v}=2\mathbf{i}-\mathbf{j}+5\mathbf{k}\) and \(\ \mathbf{w}=3\mathbf{i}+\mathbf{j}-\mathbf{k}\) are orthogonal, since \[ \mathbf{v}\,{\cdot}\, \mathbf{w}=6-1-5=0 \]
Since \(\mathbf{i}\,{\cdot}\, \mathbf{j}=1\,{\cdot}\, 0+ 0\,{\cdot}\, 1=0\), the standard basis vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane are orthogonal. The standard basis vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) in space are mutually orthogonal, since \(\mathbf{i}\,{\cdot}\, \mathbf{j}=0,\) \(\mathbf{j}\,{\cdot}\, \mathbf{k}=0,\) and \(\mathbf{k}\,{\cdot}\, \mathbf{i}=0.\)