# What is the projection of < -2, 4, 7> onto < -5, 9, -5>?

Jul 22, 2018

The projection is $= \frac{11}{131} \cdot < - 5 , 9 , - 5 >$

#### Explanation:

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

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| \vec{a} {|}^{2}} \vec{a}$

Here,

$\vec{a} = < - 5 , 9 , - 5 >$

$\vec{b} = < - 2 , 4 , 7 >$

Therefore,

The dot product is

$\vec{a} . \vec{b} = < - 5 , 9 , - 5 > . < - 2 , 4 , 7 > = 10 + 36 - 35 = 11$

The modulus of $\vec{a}$ is

$| \vec{a} | = | < - 5 , 9 , - 5 > | = \sqrt{25 + 81 + 25} = \sqrt{131}$

Therefore

$p r o {j}_{\vec{a}} \vec{b} = \frac{11}{131} \cdot < - 5 , 9 , - 5 >$