# What is the projection of (3i + 2j - 6k) onto  (-2i- 3j + 2k)?

Nov 6, 2017

The projection is $= < \frac{48}{17} , \frac{72}{17} , - \frac{48}{17} >$

#### Explanation:

Let $\vec{b} = < 3 , 2 , - 6 >$ and $\vec{a} = < - 2 , - 3 , 2 >$

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

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

$\vec{a} . \vec{b} = < - 2 , - 3 , 2 > . < 3 , 2 , - 6 > = \left(- 2\right) \cdot \left(3\right) + \left(- 3\right) \cdot \left(2\right) + \left(2\right) \cdot \left(- 6\right) = - 6 - 6 - 12 = - 24$

$| | \vec{a} | | = | | < - 2 , - 3 , 2 > | | = \sqrt{{\left(- 2\right)}^{2} + {\left(- 3\right)}^{2} + {\left(- 2\right)}^{2}} = \sqrt{4 + 9 + 4} = \sqrt{17}$

Therefore,

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| | \vec{a} | {|}^{2}} \vec{a} = - \frac{24}{17} < - 2 , - 3 , 2 >$