# What is the projection of < -3 , 1 ,3 > onto < 8 ,5 ,-3>?

Jan 28, 2017

The vector projection is $= - \frac{2}{7} < 8 , 5 , - 3 >$
The scalar projection is $= - \frac{4}{\sqrt{2}}$

#### Explanation:

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

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

The dot product is $\vec{a} . \vec{b} = < - 3 , 1 , 3 > . < 8 , 5 , - 3 >$

$= - 24 + 5 - 9 = - 28$

The modulus of $\vec{a}$ is $| | \vec{a} | | = | | < 8 , 5 , - 3 > | |$

$= \sqrt{64 + 25 + 9} = \sqrt{98}$

The vector projection is $= - \frac{28}{98} < 8 , 5 , - 3 >$

$= - \frac{2}{7} < 8 , 5 , - 3 >$

The scalar projection is $= \frac{\vec{a} . \vec{b}}{| | \vec{a} | |}$

$= - \frac{28}{\sqrt{98}} = - \frac{4}{\sqrt{2}}$