Question

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

Answer 1

The projection of a vector a onto vector b is given by

`proj_a b=(a*b)/absa^2 *a`

Hence

The dot product of `a=(-1,1,1)` and `b=(1,-1,1)` is

`a*b=-1-1+1=-1`

The magnitude of a is `absa=sqrt(-1^2+1^2+1^2)=sqrt3`

Hence the projection is

`proj_a b=-1/3*(-1,1,1)=(-1/3,1/3,1/3)=1/3*(-i+j+k)`