# What is the projection of < -3 , 1 ,3 > onto < 4, 9, 1>?

Dec 8, 2017

The vectors are perpendicular, no projection is possible.

#### 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,

$\vec{v} = < - 3 , 1 , 3 >$ and

$\vec{u} = < 4 , 9 , 1 >$

The dot product is

$\vec{u} . \vec{v} = < - 3 , 1 , 3 > . < 4 , 9 , 1 >$

$= \left(4 \cdot - 3\right) + \left(9 \cdot 1\right) + \left(3 \cdot 1\right) = - 12 + 9 + 3 = 0$

As the dot product is $= 0$, the $2$ vectors $\vec{u}$ and $\vec{v}$ are perpendicular, there is no projection.