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^{→} = (v_{1},v_{2},..., v_{n}) • (v_{1},v_{2},....,v_{n})
= v^{2}_{1}, v^{2}_{2} +.... + v^{2}_{n}
= 0
Now, since we know v^{2}_{i} ≥ 0 for all i then the only way for this sum to be zero is to in fact have v^{2}_{i} = 0. This in turn however means that we must have v_{i} = 0 and so we must have had v^{→} = 0^{→}.