Answer
$a\times (b+c)=a \times b+a \times c$
Work Step by Step
Let $a= a_1i+a_2j+a_3k$; $b=b_1i+b_2j+b_3k$ and $c=c_1i+c_2j+c_3k$
$a\times (b+c)=\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\b_1+c_1&b_2+c_2&b_3+c_3\end{vmatrix}$
Using properties of determinants, we can write
$a\times (b+c)= \begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}+\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\c_1&c_2&c_3\end{vmatrix}$
But,
$ \begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}=a \times b$
$\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\c_1&c_2&c_3\end{vmatrix}=a \times c$
Thus, $\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}+\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\c_1&c_2&c_3\end{vmatrix}=a \times b+a \times c$
Hence, $a\times (b+c)=a \times b+a \times c$
