Let vectors \(\textbf{a}\), \(\textbf{b}\) and \(\textbf{u}\) be defined as follows:
$$ \textbf{a} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \quad \textbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \textbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}$$
The sum \(\textbf{a}\) and \(\textbf{b}\) is:
$$ \textbf{a} + \textbf{b} = \begin{bmatrix} a_1 + b_1 \\ a_2 +b_2 \\ a_3 + b_3 \end{bmatrix} $$
This means:
$$\begin{align} \textbf{u} \times (\textbf{a} + \textbf{b}) &= \langle u_2(a_3 + b_3) - u_3(a_2 +b_2), -(u_1(a_3 + b_3)-u_3(a_1 + b_1) ), u_1(a_2 + b_2) - u_2(a_1 + b_1) \rangle \\ &= \langle u_2a_3 + u_2b_3 - u_3a_2 - u_3b_2, -u_1a_3 - u_1b_3+u_3a_1 + u_3b_1, u_1a_2 + u_1b_2 - u_2a_1 -u_2 b_1 \rangle \\ &= \langle (u_2a_3 - u_3a_2) + (u_2b_3 - u_3b_2), -(u_1a_3 - u_3a_1) - (u_1b_3 - u_3b_1), (u_1a_2- u_2a_1) + (u_1b_2 - u_2 b_1) \rangle\end{align}$$
According to the sum property of vector:
$$\begin{gather} \textbf{u} \times (\textbf{a} + \textbf{b}) = \langle u_2a_3 - u_3a_2, -(u_1a_3 - u_3a_1), u_1a_2- u_2a_1 \rangle +\langle u_2b_3 - u_3b_2, - (u_1b_3 - u_3b_1), u_1b_2 - u_2 b_1 \rangle \\ \therefore \textbf{u} \times (\textbf{a} + \textbf{b}) = (\textbf{u} \times \textbf{a}) + (\textbf{u} \times \textbf{b}) \end{gather}$$