# What is the projection of < -3, 6, 7> onto < 5, -3, 8>?

Feb 5, 2017

The vector projection is $= \frac{53}{98} < 5 , - 3 , 8 >$
The scalar projection is$= \frac{53}{\sqrt{98}}$

#### 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

$\vec{a} . \vec{b} = < 3 , 6 , 7 > . < 5 , - 3 , 8 \ge 15 - 18 + 56 = 53$

The modulus of $\vec{a}$ is

$| | \vec{a} | | = | | < 5 , - 3 , 8 > | | = \sqrt{25 + 9 + 64} = \sqrt{98}$

The vector projection is

$= \frac{53}{98} < 5 , - 3 , 8 >$

The scalar projection is

$\frac{\vec{a} . \vec{b}}{| | \vec{a} | |} = \frac{53}{\sqrt{98}}$