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Isn't it dot product which finds the area?

Cross product most likely, but forget about the result being a vector.

Oh, that's right, cross product because it is a*b*sin(Θ).

b*sin(Θ) is the height of the parallelogram. Duh, should have remembered the basic trig on that one.

Is there any way to solve for the area without trig?

Not really...

I suppose you could try to use pythagoras

^ what do you mean? By stacking up right triangles and using subtraction?

No- there must be an elegant way of finding B perp A

as a function of Ax, Bx, Ay and By

I'd rather just do the cross-product.