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

1. Apr 4, 2015

### Seaborgium

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
a x (b x c) = b (ac) - c (ab)

3. The attempt at a solution
I've applied the triple product expansion and reached

A x (B x A) = B (AA) - A (AB) = |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 (AB) , I can rearrange to find

|A|2 B = A x (B x A) + A (AB)

Dividing through by |A|2 results in

B = A x (B x A) |A|-2 + A ((AB)/|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!

Last edited: Apr 4, 2015
2. Apr 5, 2015

### 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.