Algebraic Properties of Dot Product

Theorem

Let

• $\mathbf{u},\mathbf{v},\mathbf{w}$ : Vectors in $R^n$
• $k$ : Any scalar

Then

• $\mathbf{u}\cdot \mathbf{v}=\mathbf{v}\cdot \mathbf{u}$ (Symmetric property)
• $\mathbf{u}\cdot (\mathbf{v}+\mathbf{w})=\mathbf{u}\cdot \mathbf{v}+\mathbf{u}\cdot \mathbf{w}$ (Distributive property)
• $k(\mathbf{u}\cdot \mathbf{v})=(k\mathbf{u})\cdot \mathbf{v}$ (Homogeneity property)
• $\mathbf{v}\cdot \mathbf{v}\ge 0$. $\mathbf{v}\cdot \mathbf{v}=0⟺\mathbf{v}=0$ (Positivity property)