Question

What is the projection of `(-i + j + k)` onto `( i - j + k)`?

Answer 1

The projection is `[(-1)/(3)][hati-hatj+hatk]`.

Explanation:

Let us assume `veca=(-hati+hatj+hatk)` and `vecb=(hati-hatj+hatk)`.
The projection of vector '`vecb`' on '`veca`' `=[(vecb.veca)/(|veca|^2)][veca]`.
`veca.vecb=a_1b_1+a_2b_2+a_3b_3=-1`.
`|veca|=sqrt[(-1)^2+(1)^2+(1)^2}=sqrt(3)`.
Substitute the above values in the projection equation we get,
`:.` projection of`vecb` on `veca` is `[(-1)/(sqrt(3))^2][hati-hatj+hatk]=(-1/3)(hati-hatj+hatk)`.