# What is the projection of < 5 , -6, 0 > onto < 3 , -7, 4 >?

Jul 17, 2016

$\left(- \frac{81}{\sqrt{74}} , - \frac{189}{\sqrt{74}} , \frac{108}{\sqrt{74}}\right)$.

#### Explanation:

Let $\vec{x} = \left(5 , - 6 , 0\right)$ and $\vec{y} = \left(3 , - 7 , 4\right)$

Then the Projection Vector of $\vec{x}$ on $\vec{y}$, denoted by, $P r o {j}_{\vec{y}} \vec{x}$, and is defined by,

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

$= \frac{\left(5 , 6 , 0\right) . \left(3 , - 7 , 4\right)}{\sqrt{{3}^{2} + {\left(- 7\right)}^{2} + {4}^{2}}} \left(3 , - 7 , 4\right)$

$= \left\{\frac{15 - 42}{\sqrt{74}}\right\} \left(3 , - 7 , 4\right)$

$= \left(- \frac{27}{\sqrt{74}}\right) \left(3 , - 7 , 4\right)$

$= \left(- \frac{81}{\sqrt{74}} , - \frac{189}{\sqrt{74}} , \frac{108}{\sqrt{74}}\right)$.