# What is the projection of < 5 ,- 3, 2 > onto < 3 , -6, 7 >?

Sep 18, 2016

$P r o {j}_{\vec{y}} \vec{x} = \left(\frac{3}{2} , - 3 , \frac{7}{2}\right)$.

#### Explanation:

The Projection Vector of $\vec{x} \ne \vec{0}$ onto $\vec{y} \ne \vec{0}$ is

denoted by, $P r o {j}_{\vec{y}} \vec{x}$, and, is defined by,

$P r o {j}_{\vec{y}} \vec{x} = \left\{\frac{\vec{x} \cdot \vec{y}}{|} | y | {|}^{2}\right\} \vec{y}$, where,

vecx"cancel(bot) vecy.

Here, $\vec{x} = \left(5 , - 3 , 2\right) \mathmr{and} \vec{y} = \left(3 , - 6 , 7\right)$.

$\therefore \vec{x} \cdot \vec{y} = 5 \cdot 3 + \left(- 3\right) \left(- 6\right) + 2 \cdot 7 = 47$, and,

$| | \vec{y} | {|}^{2} = 9 + 36 + 49 = 94$.

Hence,

$P r o {j}_{\vec{y}} \vec{x} = \frac{47}{94} \left(3 , - 6 , 7\right) = \frac{1}{2} \left(3 , - 6 , 7\right)$. Thus,

$P r o {j}_{\vec{y}} \vec{x} = \left(\frac{3}{2} , - 3 , \frac{7}{2}\right)$.

Enjoy Maths.!