(edit:solved) Vector Triple Product, Components Parallel and Perpendicular

[Seaborgium]

Homework Statement

By considering A x (B x A) resolve vector B into a component parallel to a given vector A and a component perpendicular to a given vector A.

Homework Equations

a x (b x c) = b (a ⋅ c) - c (a ⋅ b)

The Attempt at a Solution

I've applied the triple product expansion and reached

A x (B x A) = B (A ⋅ A) - A (A ⋅ B) = |A|² B - |A||B|cos(θ) A

and hit a brick wall. I'm not entirely sure what the question is asking me to do, and I feel like I'm missing crucial information. Should I be splitting vectors A and B into their cartesian components?EDIT:

I got it. Taking A x (B x A) = |A|² B - A (A ⋅ B) , I can rearrange to find

|A|² B = A x (B x A) + A (A ⋅ B)

Dividing through by |A|² results in

B = A x (B x A) |A|^-2 + A ((A ⋅ B)/|A|²)

So B is given as a component perpendicular (first term) and parallel (second term) to A.

I'm fairly sure this is right anyway. Stupid I suddenly work this out after posting here after looking at it for 30 mins before!

 

[FactChecker]

FYI, The second term is called the "projection" of B on A and the first term is called the "rejection" of B on A.