# What is the projection of <4,-6,3 > onto <1,5,2 >?

Jun 4, 2018

The vector projection is $= < - \frac{2}{3} , - \frac{10}{3} , - \frac{4}{3} >$

#### Explanation:

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

$p r o {j}_{\vec{u}} \left(\vec{v}\right) = \frac{< \vec{u} , \vec{v} >}{< \vec{u} , \vec{u} >} \vec{u}$

$\vec{u} = < 1 , 5 , 2 >$

$\vec{v} = < 4 , - 6 , 3 >$

The dot product is

$< \vec{u} , \vec{v} > = < 1 , 5 , 2 > . < 4 , - 6 , 3 >$

$= \left(1 \times 4\right) + \left(5 \times - 6\right) + \left(2 \times 3\right)$

$= 4 - 30 + 6$

$= - 20$

The magnitude of $\vec{u}$ is

$< \vec{u} , \vec{u} > = | | < 1 , 5 , 2 > | | = \sqrt{{1}^{2} + {\left(5\right)}^{2} + {2}^{2}}$

$= \sqrt{1 + 25 + 4}$

$= \sqrt{30}$

Therefore, the vector projection is

$p r o {j}_{\vec{u}} \left(\vec{v}\right) = - \frac{20}{30} < 1 , 5 , 2 >$

$= < - \frac{2}{3} , - \frac{10}{3} , - \frac{4}{3} >$