# What is the projection of  (-9 i + j + 2 k) onto (14i - 7j - 7k)?

Feb 15, 2016

$p r o {j}_{\vec{B}} \vec{A} = < - 7 i + 3 , 5 j + 3 , 5 k >$

#### Explanation:

$\vec{A} = < - 9 i + j + 2 k >$
$\vec{B} = < 14 i - 7 j - 7 k >$
$p r o {j}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{|} | \vec{B} | {|}^{2} \cdot \vec{B}$
$\vec{A} \cdot \vec{B} = - 9 \cdot 14 - 1 \cdot 7 - 2 \cdot 7$
$\vec{A} \cdot \vec{B} = - 126 - 7 - 14 = - 147$
$| | \vec{B} | {|}^{2} = {14}^{2} + {\left(- 7\right)}^{2} + {\left(- 7\right)}^{2} = 196 + 49 + 49 = 294$
$p r o {j}_{\vec{B}} \vec{A} = - \frac{147}{294} \cdot < 14 i - 7 j - 7 k >$
$p r o {j}_{\vec{B}} \vec{A} = < - 7 i + 3 , 5 j + 3 , 5 k >$