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

Jul 29, 2017

No projection since the vectors are perpendicular.

#### Explanation:

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

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

$= \frac{\vec{a} . \vec{b}}{| | \vec{a} | {|}^{2}} \cdot \vec{a}$

The dot product is

$\vec{a} . \vec{b} = < - 1 , 1 , 1 > . < 1 , - 2 , 3 > = \left(- 1 \cdot 1\right) + \left(1 \cdot - 2\right) + \left(1 \cdot 3\right)$

$= - 1 - 2 + 3 = 0$

The vectors $\vec{a}$ and $\vec{b}$ are perpendicular.

So there is no projection posiible.