Let \(\mathbf{a},\ \mathbf{b} \) and \(\mathbf{c} \) be are vectors in \(\mathbb{R}^{n}\). Then, the dot product of the vectors has the following property:
$$\mathbf{a}\cdot (\mathbf{b}+ \mathbf{c})=\mathbf{a}\cdot \mathbf{b}+\mathbf{a}\cdot \mathbf{c}$$
Proof
\begin{eqnarray*}\mathbf{a}\cdot (\mathbf{b}+\mathbf{c})&=&[a_{1} \ a_{2}\ \cdots\ a_{n}]\begin{bmatrix}b_{1}+c_{1} \\ b_{2}+c_{2}\\\vdots\\ b_{n}+c_{n}\\\end{bmatrix}\\ && \\ &=&a_{1}(b_{1}+c_{1})+a_{2}(b_{2}+c_{2})+a_{3}(b_{3}+c_{3})\\ && \\&=&(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})+(a_{1}c_{1}+a_{2}c_{2}+a_{3}c_{3})\\&& \\&=&\mathbf{a}\cdot \mathbf{b}+\mathbf{a}\cdot \mathbf{c}\end{eqnarray*}