Answer
(a) ${\bf{v}} = {{\bf{v}}_{||{\bf{w}}}} + {{\bf{v}}_{ \bot {\bf{w}}}} = \left( {\frac{{12}}{7}, - \frac{6}{7},\frac{3}{7}} \right) + \left( { - \frac{5}{7}, - \frac{1}{7},\frac{{18}}{7}} \right)$
(b) ${\bf{w}} = {{\bf{w}}_{||{\bf{v}}}} + {{\bf{w}}_{ \bot {\bf{v}}}} = \left( {\frac{9}{{11}}, - \frac{9}{{11}},\frac{{27}}{{11}}} \right) + \left( {\frac{{35}}{{11}}, - \frac{{13}}{{11}}, - \frac{{16}}{{11}}} \right)$
Work Step by Step
Recall from Theorem 3 of Section 13.3, the projection of ${\bf{v}}$ along ${\bf{w}}$ is given by Eq. (4):
${{\bf{v}}_{||{\bf{w}}}} = \left( {\frac{{{\bf{v}}\cdot{\bf{w}}}}{{{\bf{w}}\cdot{\bf{w}}}}} \right){\bf{w}}$
A vector ${{\bf{v}}_{ \bot {\bf{w}}}}$ that is orthogonal to ${\bf{w}}$ is given by
${{\bf{v}}_{ \bot {\bf{w}}}} = {\bf{v}} - {{\bf{v}}_{||{\bf{w}}}}$
Similarly, the projection of ${\bf{w}}$ along ${\bf{v}}$ is given by
${{\bf{w}}_{||{\bf{v}}}} = \left( {\frac{{{\bf{w}}\cdot{\bf{v}}}}{{{\bf{v}}\cdot{\bf{v}}}}} \right){\bf{v}}$
A vector ${{\bf{w}}_{ \bot {\bf{v}}}}$ that is orthogonal to ${\bf{v}}$ is given by
${{\bf{w}}_{ \bot {\bf{v}}}} = {\bf{w}} - {{\bf{w}}_{||{\bf{v}}}}$
(a) We have ${\bf{v}} = \left( {1, - 1,3} \right)$ and ${\bf{w}} = \left( {4, - 2,1} \right)$.
${{\bf{v}}_{||{\bf{w}}}} = \left( {\frac{{{\bf{v}}\cdot{\bf{w}}}}{{{\bf{w}}\cdot{\bf{w}}}}} \right){\bf{w}}$
${{\bf{v}}_{||{\bf{w}}}} = \left( {\frac{{\left( {1, - 1,3} \right)\cdot\left( {4, - 2,1} \right)}}{{\left( {4, - 2,1} \right)\cdot\left( {4, - 2,1} \right)}}} \right)\left( {4, - 2,1} \right)$
${{\bf{v}}_{||{\bf{w}}}} = \left( {\frac{9}{{21}}} \right)\left( {4, - 2,1} \right) = \left( {\frac{{12}}{7}, - \frac{6}{7},\frac{3}{7}} \right)$
Evaluate ${{\bf{v}}_{ \bot {\bf{w}}}}$.
${{\bf{v}}_{ \bot {\bf{w}}}} = {\bf{v}} - {{\bf{v}}_{||{\bf{w}}}}$
${{\bf{v}}_{ \bot {\bf{w}}}} = \left( {1, - 1,3} \right) - \left( {\frac{{12}}{7}, - \frac{6}{7},\frac{3}{7}} \right) = \left( { - \frac{5}{7}, - \frac{1}{7},\frac{{18}}{7}} \right)$
So,
${\bf{v}} = {{\bf{v}}_{||{\bf{w}}}} + {{\bf{v}}_{ \bot {\bf{w}}}} = \left( {\frac{{12}}{7}, - \frac{6}{7},\frac{3}{7}} \right) + \left( { - \frac{5}{7}, - \frac{1}{7},\frac{{18}}{7}} \right)$
(b) We have ${\bf{v}} = \left( {1, - 1,3} \right)$ and ${\bf{w}} = \left( {4, - 2,1} \right)$.
${{\bf{w}}_{||{\bf{v}}}} = \left( {\frac{{{\bf{w}}\cdot{\bf{v}}}}{{{\bf{v}}\cdot{\bf{v}}}}} \right){\bf{v}}$
${{\bf{w}}_{||{\bf{v}}}} = \left( {\frac{{\left( {4, - 2,1} \right)\cdot\left( {1, - 1,3} \right)}}{{\left( {1, - 1,3} \right)\cdot\left( {1, - 1,3} \right)}}} \right)\left( {1, - 1,3} \right)$
${{\bf{w}}_{||{\bf{v}}}} = \left( {\frac{9}{{11}}} \right)\left( {1, - 1,3} \right) = \left( {\frac{9}{{11}}, - \frac{9}{{11}},\frac{{27}}{{11}}} \right)$
Evaluate ${{\bf{w}}_{ \bot {\bf{v}}}}$.
${{\bf{w}}_{ \bot {\bf{v}}}} = {\bf{w}} - {{\bf{w}}_{||{\bf{v}}}}$
${{\bf{w}}_{ \bot {\bf{v}}}} = \left( {4, - 2,1} \right) - \left( {\frac{9}{{11}}, - \frac{9}{{11}},\frac{{27}}{{11}}} \right) = \left( {\frac{{35}}{{11}}, - \frac{{13}}{{11}}, - \frac{{16}}{{11}}} \right)$
So,
${\bf{w}} = {{\bf{w}}_{||{\bf{v}}}} + {{\bf{w}}_{ \bot {\bf{v}}}} = \left( {\frac{9}{{11}}, - \frac{9}{{11}},\frac{{27}}{{11}}} \right) + \left( {\frac{{35}}{{11}}, - \frac{{13}}{{11}}, - \frac{{16}}{{11}}} \right)$
