# What is the projection of <-3,5,-9 > onto <2,-1,4 >?

Feb 14, 2017

The vector projection is $= - \frac{47}{21} < 2 , - 1 , 4 >$
The scalar projection is $= - \frac{47}{\sqrt{21}}$

#### Explanation:

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

$= \frac{\vec{a} . \vec{b}}{| \vec{a} {|}^{2}} \cdot \vec{a}$

The dot product is

$\vec{a} . \vec{b} = < 2 , - 1 , 4 > . < - 3 , 5 , - 9 \ge - 6 - 5 - 36 = - 47$

The modulus of $\vec{a}$ is

$= | | < 2 , - 1 , 4 > | | = \sqrt{4 + 1 + 16} = \sqrt{21}$

The vector projection is

$= - \frac{47}{21} < 2 , - 1 , 4 >$

The scalar projection is

$= \frac{\vec{a} . \vec{b}}{|} | \vec{a} | | = - \frac{47}{\sqrt{21}}$