Answer
See the explanation below.
Work Step by Step
Vectors are $u={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j},v={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$, and $w={{a}_{3}}\mathbf{i}+{{b}_{3}}\mathbf{j}$. Now, put these vectors in the expression $c\left( u+v \right)=cu+cv$. Then, we get
$\begin{align}
& c\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}+{{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right)=c\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)+c\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right) \\
& c\left[ \left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \right]=c{{a}_{1}}\mathbf{i}+c{{b}_{1}}\mathbf{j}+c{{a}_{2}}\mathbf{i}+c{{b}_{2}}\mathbf{j} \\
& c{{a}_{1}}\mathbf{i}+c{{a}_{2}}\mathbf{i}+c{{b}_{1}}\mathbf{i}+c{{b}_{2}}\mathbf{j}=c{{a}_{1}}\mathbf{i}+c{{a}_{2}}\mathbf{i}+c{{b}_{1}}\mathbf{j}+c{{b}_{2}}\mathbf{j}
\end{align}$
Left side and right side are equal. Hence, $c\left( u+v \right)=cu+cv$.
