Nevermind, I made a careless error and that last line should have all magnitudes squared.
||b||a + ||a||b and ||b||a - ||a||b
Here's my attempt:
Since I wish to show that these vectors are orthogonal, I will attempt to show that the dot product must be 0.
(||b||a + ||a||b) * (||b||a - ||a||b)
= ||b||(a * a) + (||b||a * ||a||b) + (||a||b * ||b||a) + ||a||(b * -b)
= ||b||||a||^2 + ||b||^2||a||^2 - ||b||^2||a||^2 + ||a||||b||^2(-1)
= ||b||||a||^2 - ||a||||b||^2
But I see nothing to suggest that this is 0.
Hints?