Dot Product Associates with Scalar Multiplication

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Theorem

Let $\mathbf u, \mathbf v$ be vectors in the real Euclidean space $\R^n$.

Let $c$ be a real scalar.


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