FP3 cross product proof of the direction

I've seen a proof of 2 vectors a,b laxbl = lallblsin(angle between them)

https://math.la.asu.edu/~surgent/mat272/cross_mag_proof.pdf

but I can't find a proof on google of why the direction of the vector from the cross product of a and b is perpendicular to both of them?

anyone got any good sites or videos, or can actually post their own proof? thanks

There's no proof. It's simply the part of the definition of the cross product. It's defined that way essentially because there are physical vector quantities that combine in such as way as to produce a vector that is most usefully shown as perpendicular to the plane of the other two e.g.

a) the Lorentz force F on a moving charged particle in a magnetic field B is perpendicular to the plane containing B and the velocity vector v of the particle.

b) if we draw a torque vector as perpendicular to the plane containing the force and displacement vectors, then we find that the torque vectors add together correctly when we have multiple forces turning an object.

So it's defined to point in a perpendicular direction since that happens to model some physical situations nicely.