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
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $2$. The Scalar Product: $(2.1)$
- 1992: Frederick W. Byron, Jr. and Robert W. Fuller: Mathematics of Classical and Quantum Physics ... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product