# What is the projection of <7,-5,6 > onto <-1,-3,7 >?

##### 1 Answer
Feb 13, 2017

The vector projection is $= \frac{50}{59} < - 1 , - 3 , 7 >$
The scalar projection is $= \frac{50}{\sqrt{59}}$

#### 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

$\vec{a} . \vec{b} = < - 1 , - 3 , 7 > . < 7 , - 5 , 6 > = - 7 + 15 + 42 = 50$

The modulus of $\vec{a}$ is

$| | < - 1 , - 3 , 7 > | | = \sqrt{1 + 9 + 49} = \sqrt{59}$

The vector projection is $= \frac{50}{59} < - 1 , - 3 , 7 >$

The scalar projection is $= \frac{\vec{a} . \vec{b}}{| | \vec{a} | |} = \frac{50}{\sqrt{59}}$