# What is the projection of <4 , -6, 7 > onto < -1 , 9,-2 >?

Jul 22, 2017

The vector projection is $= - \frac{72}{86} < - 1 , 9 , - 2 >$
The scalar projection is $= - \frac{72}{\sqrt{86}}$

#### Explanation:

Let $\vec{b} = < 4 , - 6 , 7 >$ and $\vec{a} = < - 1 , 9 , - 2 >$

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

$= \frac{\vec{a} . \vec{b}}{| | \vec{a} | {|}^{2}} \cdot \vec{a}$

The dot product is

$\vec{a} . \vec{b} = < 4 , - 6 , 7 > . < - 1 , 9 , - 2 > = \left(4 \cdot - 1\right) + \left(- 6 \cdot 9\right) + \left(7 \cdot - 2\right)$

$= - 4 - 54 - 14 = - 72$

The modulus of $\vec{a}$ is

$| | < - 1 , 9 , - 2 > | | = \sqrt{1 + 81 + 4} = \sqrt{86}$

Therefore,

The vector projection is

$= - \frac{72}{86} < - 1 , 9 , - 2 >$

The scalar projection is

$= \frac{\vec{a} . \vec{b}}{| | \vec{a} | |} = - \frac{72}{\sqrt{86}}$