Decomposing a Vector into Two Orthogonal Vectors

Find the vector projection of \(\mathbf{v}=2\mathbf{i}-\mathbf{j}+\mathbf{k}\) onto \(\mathbf{w}=\mathbf{i}+\mathbf{j}+\mathbf{k}\). Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v} _{1} \) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \( \mathbf{w}\).

Solution We use the formula for the projection of \(\mathbf{v}\) onto \(\mathbf{w}.\) \[ \begin{eqnarray*} \mathbf{v}_{1}& =&\hbox{proj}_{\mathbf{w}}\mathbf{v}=\frac{\mathbf{v}\,{\cdot}\, \mathbf{w}}{\Vert \mathbf{w}\Vert ^{2}}\mathbf{w} \underset{\underset{\underset{{\color{#0066A7}{\hbox{\(\Vert \mathbf{w}\Vert =\sqrt{1^{2}+1^{2} +1^{2}} =\sqrt{3}\)}}}} {\color{#0066A7}{\hbox{\(\mathbf{v}\,{\cdot}\, \mathbf{w=}2-1+1=2\)}}}} {\color{#0066A7}{\uparrow }}}{=}\frac{2}{(\sqrt{3})^{2}}\mathbf{w}=\frac{2}{3}( \mathbf{i}+\mathbf{j}+\mathbf{k})=\frac{2}{3}\,\mathbf{i}+\frac{2}{3}\, \mathbf{j}+\frac{2}{3}\,\mathbf{k} \\ \mathbf{v}_{2}& =&\mathbf{v}-\mathbf{v}_{1}=(2\mathbf{i}-\mathbf{j}+\mathbf{k} )-\left( \frac{2}{3}\,\mathbf{i}+\frac{2}{3}\,\mathbf{j}+\frac{2}{3}\, \mathbf{k}\right) =\frac{4}{3}\,\mathbf{i}-\frac{5}{3}\,\mathbf{j}+\frac{1}{3 }\,\mathbf{k} \end{eqnarray*} \]