# 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 |

$\blacksquare$