Dot Product Associates with Scalar Multiplication
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Theorem
Let $\mathbf u, \mathbf v$ be vectors in the real Euclidean space $\R^n$.
Then:
- $\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$
Proof 1
\(\ds \left({c \mathbf u}\right) \cdot \mathbf v\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \left({c u_i}\right) v_i\) | Definition of Dot Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n c \left({ u_i v_i }\right)\) | Real Multiplication is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds c \sum_{i \mathop = 1}^n u_i v_i\) | Real Multiplication Distributes over Real Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds c \left({\mathbf u \cdot \mathbf v}\right)\) | Definition of Dot Product |
$\blacksquare$
Proof 2
\(\ds \paren {c \mathbf u} \cdot \mathbf v\) | \(=\) | \(\ds \norm {c \mathbf u} \norm {\mathbf v} \cos \angle c \mathbf u, \mathbf v\) | 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 |
$\blacksquare$
Proof 3
From Dot Product Operator is Bilinear:
- $\left({c \mathbf u + \mathbf v}\right) \cdot \mathbf w = c \left({\mathbf u \cdot \mathbf w}\right) + \left({\mathbf v \cdot \mathbf w}\right)$
Setting $\mathbf v = 0$ and renaming $\mathbf w$ yields the result.
$\blacksquare$
Sources
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 2$.