# What is the projection of (i -2j + 3k) onto  ( 2i+j+2k)?

Jul 8, 2018

The projection is $= < 1 , \frac{1}{2} , \frac{1}{2} >$

#### Explanation:

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

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

The vectors are

$\vec{a} = < 2 , 1 , 1 >$

and

$\vec{b} = < 1 , - 2 , 3 >$

The dot product is

$\vec{a} . \vec{b} = < 2 , 1 , 1 > . < 1 , - 2 , 3 >$

$= \left(2 \times 1\right) + \left(1 \times \left(- 2\right)\right) + 1 \times 3 = 2 - 2 + 3 = 3$

The magnitude of $\vec{a}$ is

$| | \vec{a} | | = < 2 , 1 , 1 > = \sqrt{{2}^{2} + {1}^{2} + {1}^{2}} = \sqrt{6}$

Therefore,

$p r o {j}_{\vec{a}} \vec{b} = \frac{3}{6} \cdot < 2 , 1 , 1 >$

$= < 1 , \frac{1}{2} , \frac{1}{2} >$