Question

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

Answer 1

`\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 `\vec{A_{}}` onto another vector `\vec{B_{}}` is:

`\vec{A_B} = \frac{\vec{A_{}}.\vec{B_{}}}{B}\hat{B}=\frac{\vec{A_{}}.\vec{B_{}}}{B^2}\vec{B_{}}`

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

`\vec{A_{}}.\vec{B_{}}=(2\times1-4\times2+3\times2)=0`

Since `\vec{A_{}}.\vec{B_{}} = 0`, it is clear that `\vec{A_{}}` is perpendicular to `\vec{B_{}}`. So its projection on `\vec{B_{}}` is a null vector.