Question

What is the angle between A+B = A-B?

Answer 1

Let A and B be vectors OP and OQ respectively.  

A and B vectors form a parallelogram in the two dimensional space.  Then A+B and A-B are diagonals OR and QP.   The angle between them is the angle QSO between the diagonals at intersection S in the triangle OQS.

Let angle between vectors A and B be θ.

Angle between B (vector OQ) and vector A+B
$=\angle{SOQ}=tan^{-1}\frac{|A|\ Sin\theta}{|B|+|A| Cos\theta}\\\\Angle\ between \vector{B}\ and\ \vec{A-B}=\angle{OQS} =tan^{-1}\frac{|A|\ Sin\theta}{|B|-|A| Cos\theta}\\\\In \triangle OQS, \angle{OSQ}=angle\ betw\ \vec{A}\ and\ \vec{B}=180-\angle{SOQ}-\angle{OQS}\\\\tan\angle{OSQ}=-tan(\angle{SOQ}+\angle{OQS})\\\\=\frac{2|A|\ |B|\ Sin\theta}{|\ |A|^2-|B|^2\ |}\\\\=\frac{2| \vec{A} \times \vec{B} |}{|\ |A|^2-|B|^2\ |}$