Dot Product Operator is Commutative/Proof 2

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Theorem

$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$


Proof

\(\ds \mathbf u \cdot \mathbf v\) \(=\) \(\ds \norm {\mathbf u} \norm {\mathbf v} \cos \angle \mathbf u, \mathbf v\) Definition of Dot Product
\(\ds \) \(=\) \(\ds \norm {\mathbf v} \norm {\mathbf u} \cos \angle \mathbf u, \mathbf v\) Real Multiplication is Commutative
\(\ds \) \(=\) \(\ds \norm {\mathbf v} \norm {\mathbf u} \cos \angle \mathbf v, \mathbf u\) Cosine Function is Even
\(\ds \) \(=\) \(\ds \mathbf v \cdot \mathbf u\) Definition of Dot Product

$\blacksquare$


Sources