# What is the projection of <6,-4,9 > onto <7,8,7 >?

Feb 4, 2017

The vector projection is $= \frac{73}{162} < 7 , 8 , 7 >$
The scalar projection $= \frac{73}{\sqrt{162}}$

#### 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

$\vec{a} . \vec{b} = < 7 , 8 , 7 > . < 6 , - 4 , 9 \ge 42 - 32 + 63 = 73$

The modulus of $\vec{a}$ is

$| | \vec{a} | | = | | < 7 , 8 , 7 > | | = \sqrt{49 + 64 + 49} = \sqrt{162}$

The vector projection is

$= \frac{73}{162} < 7 , 8 , 7 >$

The scalar projection is

$\frac{\vec{a} . \vec{b}}{| | \vec{a} | |} = \frac{73}{\sqrt{162}}$