# What is the projection of < 6 , -6 ,3 > onto < 4, 9, 1>?

Jan 21, 2017

The vector projection is $= - \frac{27}{98} < 4 , 9 , 1 >$
The scalar projection is $= - \frac{27}{\sqrt{98}}$

#### 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

$\vec{a} . \vec{b} = < 6 , - 6 , 3 > . < 4 , 9 , 1 \ge 24 - 54 + 3 = - 27$

The modulus is

$| \vec{a} | = | < 4 , 9 , 1 > | = \sqrt{16 + 81 + 1} = \sqrt{98}$

The vector projection is $= - \frac{27}{98} < 4 , 9 , 1 >$

The scalar projection is

$= \vec{a} . \frac{\vec{b}}{|} \vec{a} | = - \frac{27}{\sqrt{98}}$