# What is the projection of <3 ,-4, 7 > onto < -1 , 0, -2 >?

Apr 23, 2018

Projection of $\vec{a} = < 3 , - 4 , 7 >$ onto $\vec{b} < - 1 , 0 , - 2 >$ is $< - \frac{51}{74} , \frac{34}{37} , - \frac{119}{74} >$

#### Explanation:

Projection of $\vec{a} = < {x}_{1} , {y}_{1} , {z}_{1} >$ onto $\vec{b} = < {x}_{2} , {y}_{2} , {z}_{2} >$ is $\frac{\vec{a} \cdot \vec{b}}{|} \vec{a} {|}^{2} \cdot \vec{a}$

Hence for projection of $\vec{a} = < 3 , - 4 , 7 >$ onto $\vec{b} < - 1 , 0 , - 2 >$

as $\vec{a} \cdot \vec{b} = 3 \times \left(- 1\right) + \left(- 4\right) \times 0 + 7 \times \left(- 2\right) = - 17$

$| \vec{a} {|}^{2} = {3}^{2} + {\left(- 4\right)}^{2} + {7}^{2} = 74$

Projection of $\vec{a} = < 3 , - 4 , 7 >$ onto $\vec{b} < - 1 , 0 , - 2 >$ is

$- \frac{17}{74} < 3 , - 4 , 7 \ge < - \frac{51}{74} , \frac{68}{74} , - \frac{119}{74} >$ or $< - \frac{51}{74} , \frac{34}{37} , - \frac{119}{74} >$