# What is the projection of <2,-4,3 > onto <1,2,2 >?

Feb 28, 2016

$\setminus \vec{{A}_{}}$ id perpendicular to \vec{B_{}. So one's projection on the other must be a null vector ( $< 0 , 0 , 0 >$)

#### Explanation:

The projection of a $\setminus \vec{{A}_{}}$ onto another vector $\setminus \vec{{B}_{}}$ is:

$\setminus \vec{{A}_{B}} = \setminus \frac{\setminus \vec{{A}_{}} . \setminus \vec{{B}_{}}}{B} \setminus \hat{B} = \setminus \frac{\setminus \vec{{A}_{}} . \setminus \vec{{B}_{}}}{{B}^{2}} \setminus \vec{{B}_{}}$

Solution: \vec{A_{}}=<2,-4,3>; \qquad \vec{B_{}}=<1,2,2>;

$\setminus \vec{{A}_{}} . \setminus \vec{{B}_{}} = \left(2 \setminus \times 1 - 4 \setminus \times 2 + 3 \setminus \times 2\right) = 0$

Since $\setminus \vec{{A}_{}} . \setminus \vec{{B}_{}} = 0$, it is clear that $\setminus \vec{{A}_{}}$ is perpendicular to $\setminus \vec{{B}_{}}$. So its projection on $\setminus \vec{{B}_{}}$ is a null vector.