Answer
$${\bf{w}} = \frac{{16}}{7}{\bf{i}} + \frac{{24}}{7}{\bf{j}} + \frac{8}{7}{\bf{k}}$$
Work Step by Step
$$\eqalign{
& {\text{Let }}{\bf{u}} = 5{\bf{i}} + {\bf{j}} + 3{\bf{k}},{\text{ }}{\bf{v}} = 2{\bf{i}} + 3{\bf{j}} + {\bf{k}} \cr
& {\text{Let }}{\bf{w}} = {\text{pro}}{{\text{j}}_{\bf{v}}}{\bf{u}} = \left( {\frac{{{\bf{u}} \cdot {\bf{v}}}}{{{{\left\| {\bf{v}} \right\|}^2}}}} \right){\bf{v}} \cr
& {\bf{w}} = \left( {\frac{{\left( {5{\bf{i}} + {\bf{j}} + 3{\bf{k}}} \right) \cdot \left( {2{\bf{i}} + 3{\bf{j}} + {\bf{k}}} \right)}}{{{{\left\| {2{\bf{i}} + 3{\bf{j}} + {\bf{k}}} \right\|}^2}}}} \right)\left( {2{\bf{i}} + 3{\bf{j}} + {\bf{k}}} \right) \cr
& {\bf{w}} = \left( {\frac{{10 + 3 + 3}}{{4 + 9 + 1}}} \right)\left( {2{\bf{i}} + 3{\bf{j}} + {\bf{k}}} \right) \cr
& {\bf{w}} = \frac{8}{7}\left( {2{\bf{i}} + 3{\bf{j}} + {\bf{k}}} \right) \cr
& {\bf{w}} = \frac{{16}}{7}{\bf{i}} + \frac{{24}}{7}{\bf{j}} + \frac{8}{7}{\bf{k}} \cr} $$
