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That's much neater than my algebraic solution

[(a_{x}+b_{x})^{2})+(a_{y}+b_{y})^{2})] - [(a_{x}-b_{x})^{2})**+**(a_{y}-b_{y})^{2})]=0

Ive coloured in red the right sign.

Which also ends up with the dot product (a_{x}b_{x} + a_{y}b_{y}) equals zero.

Perhaps to fully explain - the dot product of two vectors a.b = |a|.|b|CosΘ where |a| is the magnitude and Θ is the angle between the two vectors. For the dot product |a|.|b|CosΘ to equal zero either magnitude of a or b must be zero or CosΘ must be zero - and that happens when Θ is 90 or 270 degrees.

Ok. I didn't explain dot product because, in a general vectorial space, you can define it in a great variety of ways; for example, in quantum mechanics vectors are "functions" Ψ(x,y,z,t) and the dot product is defined as:

(Ψ,Φ) = |Ψ>•|Φ> = ∫∫∫Ψ*(x,y,z,t) Φ(x,y,z,t)dx dy dz

where "Ψ*" means "complex coniugate of Ψ".