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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,..., vn) • (v1,v2,....,vn)
= v21, v22 +.... + v2n
= 0
Now, since we know v2i ≥ 0 for all i then the only way for this sum to be zero is to in fact have v2i = 0. This in turn however means that we must have vi = 0 and so we must have had v→ = 0→.
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