Answer
$$\eqalign{
& \left( {\bf{a}} \right){{\bf{w}}_1} = \left\langle { - 2,2,2} \right\rangle \cr
& \left( {\bf{b}} \right){{\bf{w}}_2} = \left\langle {2,1,1} \right\rangle \cr} $$
Work Step by Step
$$\eqalign{
& {\text{Let the vectors be }}{\bf{u}} = \left\langle {0,3,3} \right\rangle ,{\text{ }}{\bf{v}} = \left\langle { - 1,1,1} \right\rangle \cr
& \left( {\bf{a}} \right){\text{ Let }}{{\bf{w}}_1} = {\text{pro}}{{\text{j}}_{\bf{v}}}{\bf{u}} = \left( {\frac{{{\bf{u}} \cdot {\bf{v}}}}{{{{\left\| {\bf{v}} \right\|}^2}}}} \right){\bf{v}} \cr
& {{\bf{w}}_1} = \left( {\frac{{\left\langle {0,3,3} \right\rangle \cdot \left\langle { - 1,1,1} \right\rangle }}{{{{\left\| {\left\langle { - 1,1,1} \right\rangle } \right\|}^2}}}} \right)\left\langle { - 1,1,1} \right\rangle \cr
& {{\bf{w}}_1} = \left( {\frac{{0 + 3 + 3}}{{1 + 1 + 1}}} \right)\left\langle { - 1,1,1} \right\rangle \cr
& {{\bf{w}}_1} = 2\left\langle { - 1,1,1} \right\rangle \cr
& {{\bf{w}}_1} = \left\langle { - 2,2,2} \right\rangle \cr
& \cr
& \left( {\bf{b}} \right){\text{From the Definitions of Projection and Vector Components}} \cr
& {{\bf{w}}_2} = {\bf{u}} - {{\bf{w}}_1}{\text{ is called the }}{\bf{vector}}{\text{ }}{\bf{component}}{\text{ }}{\bf{of}}{\text{ }}{\bf{u}}{\text{ }}{\bf{orthogonal}} \cr
& {\bf{to}}{\text{ }}{\bf{v}}. \cr
& {{\bf{w}}_2} = \left\langle {0,3,3} \right\rangle - \left\langle { - 2,2,2} \right\rangle \cr
& {{\bf{w}}_2} = \left\langle {0 + 2,3 - 2,3 - 2} \right\rangle \cr
& {{\bf{w}}_2} = \left\langle {2,1,1} \right\rangle \cr} $$
