I'm not sure about this...

Question

Find the angle formed when [3, 4] and [−5, 12] are placed tail-to-tail, then find components for the vector projection that results when [3, 4] is projected onto [−5, 12].

I figured out the angle. I'm not sure about the second part of this question. Does the projection vector travel in the perpendicular position of [-5, 12]? If so it is simply the vector [5, 12] multiplied by a scalar right? It would be much appreciated if you could show me a diagram.

Answer 1

If you are projecting vector v onto w it will travel in the direction of w, hence "onto w".

You are correct that it simply a multiplication by a scalar.

If v = <3, 4> and w = <-5, 12>, then the projection of v onto w will be v₁, where

v₁ = (v dot w)/ ||w||² w

We get:

v dot w = 3(-5) + 4(12) = 33

||w||² = (-5)² + (12)² = 169

v₁ = (v dot w)/ ||w||² w = <-165/169, 396/169>

This gives a vector in the direction of w, such that a vector v₂ (drawn from the head of v₁ to the head of v) is orthogonal to v₁.

You can see a diagram here: https://www.geogebra.org/geometry/jfkubjnw