Dot Product Associates with Scalar Multiplication/Proof 2
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Theorem
- $\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$
Proof
| \(\ds \paren {c \mathbf u} \cdot \mathbf v\) | \(=\) | \(\ds \norm {c \mathbf u} \norm {\mathbf v} \cos \angle c \mathbf u, \mathbf v\) | Definition of Dot Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\sum_{i \mathop = 1}^n \paren {c u_i}^2} \norm {\mathbf v} \cos \angle c \mathbf u, \mathbf v\) | Definition of Vector Length in $\R^n$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {c^2 \sum_{i \mathop = 1}^n u_i^2} \norm {\mathbf v} \cos \angle \mathbf u, \mathbf v\) | $c \mathbf u$ and $\mathbf u$ are in the same direction | |||||||||||
| \(\ds \) | \(=\) | \(\ds c \sqrt {\sum_{i \mathop = 1}^n u_i^2} \norm {\mathbf v} \cos \angle \mathbf u, \mathbf v\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds c \norm {\mathbf u} \norm {\mathbf v} \cos \angle \mathbf u, \mathbf v\) | Definition of Vector Length in $\R^n$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds c \paren {\mathbf u \cdot \mathbf v}\) | Definition of Dot Product |
$\blacksquare$