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

Jan 12, 2018

The projection is $= \frac{4}{23} < 1 , - 6 , - 3 >$

#### Explanation:

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

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| | \vec{a} | |} ^ 2 \vec{a}$

$\vec{a} = < 1 , - 6 , - 3 >$

$\vec{b} = < 5 , - 2 , 3 >$

The dot product is

$\vec{a} . \vec{b} = < 1 , - 6 , - 3 > . < 5 , - 2 , 3 >$

$= 1 \cdot 5 + \left(- 6\right) \cdot \left(- 2\right) + \left(- 3\right) \cdot 3 = 5 + 12 - 9 = 8$

The modulus of $\vec{a}$ is

$= | | \vec{a} | | = | | < 1 , - 6 , - 3 > | | = \sqrt{{1}^{2} + {\left(- 6\right)}^{2} + {\left(- 3\right)}^{2}} = \sqrt{46}$

Therefore,

$p r o {j}_{\vec{a}} \vec{b} = \frac{8}{46} < 1 , - 6 , - 3 > = \frac{4}{23} < 1 , - 6 , - 3 >$