Let vectors \(\textbf{v}\), \(\textbf{u}\) and \(\textbf{w}\) be defined as follows:
$$ \textbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} \quad \textbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \quad \textbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} $$
The addition \(\textbf{v} \cdot \textbf{w}\) is defined as follows:
$$ \textbf{v} + \textbf{w} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} + \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} = \begin{bmatrix} v_1 +w_1 \\ v_2+w_2 \\ v_3+w_3 \end{bmatrix}$$
The dot product \(\textbf{u} \cdot (\textbf{v} + \textbf{w})\) is defined as follows:
$$ \textbf{u} \cdot (\textbf{v} + \textbf{w}) = u_1 (v_1 +w_1) + u_2 (v_2+w_2) + u_3 (v_3+w_3) $$
This means:
$$ \begin{align} \textbf{u} \cdot (\textbf{v} + \textbf{w}) &= u_1v_1 + u_1w_1 + u_2v_2+u_2w_2 + u_3v_3+u_3w_3 \\ &=( u_1v_1 + u_2v_2 + u_3v_3)+ (u_1w_1 +u_2w_2 +u_3w_3) \end{align} $$
By definition of the dot product:
$$ (u_1v_1 + u_2v_2 + u_3v_3)+ (u_1w_1 +u_2w_2 +u_3w_3 )= (\textbf{u} \cdot \textbf{v}) + (\textbf{u} \cdot \textbf{w}) $$
Therefore, \(\textbf{u} \cdot (\textbf{v} + \textbf{w})=(\textbf{u} \cdot \textbf{v}) + (\textbf{u} \cdot \textbf{w}) \). A similar proof can be used for other dimensions.