Question

What are the algebraic properties of the dot/inner product?

Answer 1

Dot products have many of the same algebraic properties as products of real numbers. If $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ are vectors in $R^{n}$, and if $k$ is a scalar, then: $(a) \quad \mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$ (b) $\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$ (c) $\quad k(\mathbf{u} \cdot \mathbf{v})=(k \mathbf{u}) \cdot \mathbf{v}$ (d) $\mathbf{v} \cdot \mathbf{v} \geq 0$ and $\mathbf{v} \cdot \mathbf{v}=0$ if and only if $\mathbf{v}=\mathbf{0}$