Properties of Dot Product - proof

Proof of: If v^→ • v^→ = 0 then v^→ = 0^→

This is a pretty simple proof.  Let us start with v^→ = (v1 , v2 ,.... , vn) and compute the dot product.

 v^→ • v^→ = (v₁,v₂,..., v_n) • (v₁,v₂,....,v_n)

= v²₁, v²₂ +.... + v²_n

 = 0

Now, since we know v²_i ≥ 0 for all i then the only way for this sum to be zero is to in fact have v²_i = 0. This in turn however means that we must have v_i = 0 and so we must have had v^→ = 0^→.