Dot Product with Self is Zero iff Zero Vector/Proof 1

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Theorem

$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$


Proof

\(\ds \mathbf u \cdot \mathbf u\) \(=\) \(\ds 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds \sum_{i \mathop = 1}^n u_i^2\) \(=\) \(\ds 0\) Definition of Dot Product
\(\ds \leadstoandfrom \ \ \) \(\ds \forall i: \, \) \(\ds u_i\) \(=\) \(\ds 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds \mathbf u\) \(=\) \(\ds \bszero\) Definition of Zero Vector

$\blacksquare$