Orthoganality Theorem

Two non-zero vectors y and v for a right angle if and only if $⟨u,v⟩=0$ If they are right angle, we can say they are orthoganal.

Proof

Proving $⟹$

Suppose $\theta =\frac{\pi }{2}$ Then, this gives $⟨u,v⟩=\cos (\frac{\pi }{2})\mid \mid u\mid \mid v\mid \mid =0$

Proving $⟸$

Suppose $⟨u,v⟩=0$ Then, we get: $0=⟨u,v⟩=\cos (\theta )\mid \mid u\mid \mid \mid \mid v\mid \mid$ Note that $\mid \mid u\mid \mid \mid \mid v\mid \mid \ne 0$ Therefore, $\cos (\theta )=0$ we get that $\theta =\frac{\pi }{2}$ $$\square$$$$