What is the projection of < 0 , 2, -2 > onto < 3 , -6, 7 >?

Jan 1, 2017

$P r o {j}_{\vec{v}} \vec{u} = \frac{- 26}{94} < 3 , - 6 , 7 >$ =$< \frac{- 78}{94} , \frac{156}{94} , \frac{- 182}{94} >$

Explanation:

Let $\vec{u}$ be <0,2,-2> and $\vec{v}$ be <3,-6,7>. The projection of $\vec{u}$ on $\vec{v}$ is represented as

$P r o {j}_{\vec{v}} \vec{u}$ = $\frac{\vec{u} . \vec{v}}{|} | \vec{v} | {|}^{2} \vec{v}$

In the present case $\vec{u} . \vec{v}$ = 0(3) +2(-6) +(-2)(7) = -26

Also $| | v | {|}^{2}$ = ${3}^{2} + {\left(- 6\right)}^{2} + {7}^{2}$ = 94

Thus $P r o {j}_{\vec{v}} \vec{u} = \frac{- 26}{94} < 3 , - 6 , 7 >$ =$< \frac{- 78}{94} , \frac{156}{94} , \frac{- 182}{94} >$