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

$\blacksquare$