# What is the projection of <-4,8,9 > onto <8,1,-1 >?

Dec 5, 2017

The projection is $= < - 4 , - \frac{1}{2} , \frac{1}{2} >$

#### Explanation:

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

$p r o {j}_{u} v = \frac{\vec{u} . \vec{v}}{| | \vec{u} | |} ^ 2 \cdot \vec{u}$

Here,

$\vec{v} = < - 4 , 8 , 9 >$ and

$\vec{u} = < 8 , 1 , - 1 >$

The dot product is

$\vec{u} . \vec{v} = < - 4 , 8 , 9 > . < 8 , 1 , - 1 >$

$= \left(- 4 \cdot 8\right) + \left(8 \cdot 1\right) + \left(9 \cdot - 1\right) = - 32 + 8 - 9 = - 33$

The magnitude of $\vec{u}$ is

$= | | < 8 , 1 , - 1 > | | = \sqrt{{\left(8\right)}^{2} + {\left(1\right)}^{2} + {\left(- 1\right)}^{2}}$

$= \sqrt{64 + 1 + 1} = \sqrt{66}$

The vector projection is

$p r o {j}_{u} v = - \frac{33}{66} \cdot < 8 , 1 , - 1 > = < - 4 , - \frac{1}{2} , \frac{1}{2} >$