# What is the projection of (8i + 12j + 14k) onto  (2i + 3j – 7k)?

Jan 29, 2018

The vector projection is $= - \frac{36}{\sqrt{62}} < 2 , 3 , - 7 >$

#### Explanation:

The vector projection of $\vec{b}$ onto $\vec{a}$ is

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| | \vec{a} | |} ^ 2 \vec{a}$

$\vec{a} = < 2 , 3 , - 7 >$

$\vec{b} = < 8 , 12 , 14 >$

The dot product is

$\vec{a} . \vec{b} = < 2 , 3 , - 7 > . < 8 , 12 , 14 >$

$= \left(2\right) \cdot \left(8\right) + \left(3\right) \cdot \left(12\right) + \left(- 7\right) \cdot \left(14\right) = 16 + 36 - 84 = - 36$

The modulus of $\vec{a}$ is

$= | | \vec{a} | | = | | < 2 , 3 , - 7 > | | = \sqrt{{\left(2\right)}^{2} + {\left(3\right)}^{2} + {\left(- 7\right)}^{2}} = \sqrt{4 + 9 + 49} = \sqrt{62}$

Therefore,

$p r o {j}_{\vec{a}} \vec{b} = - \frac{36}{\sqrt{62}} < 2 , 3 , - 7 >$