Question

What is the projection of `< -2, -5, 7>` onto `< 7,2, -5>`?

Answer 1

`proj_(vec(b))(vec(a)) = 59/78 [(-7), (-2), (5)]`

Explanation:

Given : Vectors `vec(a)=<-2, -5, 7>` and `vecb<7,2, -5>`

Required: Projections of `vec(a)` onto `vec(b)`

Solution Strategy: `proj_(vec(b))(vec(a))=[vec(a)*vec(b)]/[vec(b)*vec(b)]*vec(b)`

`vec(a)*vec(b)=(-2*7)+(-5*2)+(7*(-5))=-59`
`vec(b)*vec(b)= 7*7+2*2+(-5*-5)`=78

`proj_(vec(b))(vec(a)) = 59/78 [(-7), (-2), (5)]`

Note: the vectors are flattened to 2D in the geometric depiction. You imager simply gives you an idea what a projection is...
Let's see what it looks like geometrically: