Answer
$${\bf{w}} = \left\langle {1,0,1} \right\rangle $$
Work Step by Step
$$\eqalign{
& {\text{Let }}{\bf{u}} = \left\langle {1, - 1,1} \right\rangle ,{\text{ }}{\bf{v}} = \left\langle {2,0,2} \right\rangle \cr
& {\text{Let }}{\bf{w}} = {\text{pro}}{{\text{j}}_{\bf{v}}}{\bf{u}} = \left( {\frac{{{\bf{u}} \cdot {\bf{v}}}}{{{{\left\| {\bf{v}} \right\|}^2}}}} \right){\bf{v}} \cr
& {\bf{w}} = \left( {\frac{{\left\langle {1, - 1,1} \right\rangle \cdot \left\langle {2,0,2} \right\rangle }}{{{{\left\| {\left\langle {2,0,2} \right\rangle } \right\|}^2}}}} \right)\left\langle {2,0,2} \right\rangle \cr
& {\bf{w}} = \left( {\frac{{2 + 0 + 2}}{{4 + 0 + 4}}} \right)\left\langle {2,0,2} \right\rangle \cr
& {\bf{w}} = \frac{1}{2}\left\langle {2,0,2} \right\rangle \cr
& {\bf{w}} = \left\langle {1,0,1} \right\rangle \cr} $$
