# What is the projection of <-7 , -7, 1 > onto < 4, 0, -2 >?

Mar 2, 2017

The vector projection is $= < - 6 , 0 , 3 >$
The scalar projection is $= - 3 \sqrt{5}$

#### Explanation:

The vector projection of $\vec{u}$ onto $\vec{v}$ is

$= \frac{\vec{u} . \vec{v}}{| | \vec{v} | {|}^{2}} \vec{v}$

The dot product is

$\vec{u} . \vec{v} = < - 7 , - 7 , 1 > , < 4 , 0 , - 2 > = - 28 + 0 - 2 = - 30$

The modulus of $\vec{v}$ is

$= | | \vec{v} | | = | | \vec{v} | | = | | < 4 , 0 , - 2 | |$

$= \sqrt{16 + 0 + 4} = \sqrt{20}$

The vector projection is

$= - \frac{30}{20} < 4 , 0 , - 2 >$

$= < - 6 , 0 , 3 >$

The scalar projection is

$= \frac{\vec{u} . \vec{v}}{| | \vec{v} | |}$

$= - \frac{30}{\sqrt{20}} = - \frac{15}{\sqrt{5}} = - 3 \sqrt{5}$