# What is the projection of (-i + j + k) onto  ( 3i + 2j - 3k)?

May 8, 2018

The projection is $= - \frac{2}{3} \vec{i} - \frac{4}{9} \vec{j} + \frac{2}{3} \vec{k}$

#### Explanation:

The vector projection of $\vec{b}$ onto $\vec{a}$ is

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| \vec{a} |} ^ 2 \vec{a}$

Here

$\vec{a} = < 3 , 2 , - 3 >$

$\vec{b} = < - 1 , 1 , 1 >$

The dot product is

$\vec{a} . \vec{b} = < 3 , 2 , - 3 > . < - 1 , 1 , 1 > = - 3 + 2 - 3 = - 4$

The maghitude of $\vec{a}$ is

$| \vec{a} | = | < 3 , 2 , - 3 > | = \sqrt{9 + 4 + 9} = \sqrt{18}$

Therefore,

$p r o {j}_{\vec{a}} \vec{b} = - \frac{4}{18} < 3 , 2 , - 3 >$

$= - \frac{2}{9} < 3 , 2 , - 3 >$

$= < - \frac{2}{3} , - \frac{4}{9} , \frac{2}{3} >$

$= - \frac{2}{3} \vec{i} - \frac{4}{9} \vec{j} + \frac{2}{3} \vec{k}$