Answer
If $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$ and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$ are vectors, the product $\mathbf{v}\cdot \mathbf{w}$, called the dot product, is defined as $\mathbf{v}\cdot \mathbf{w}={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}$.
Work Step by Step
Dot product of two vectors can be obtained as
$\begin{align}
& \mathbf{v}\cdot \mathbf{w}=\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)\cdot \left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right) \\
& ={{a}_{1}}{{a}_{2}}\left( \mathbf{i}\cdot \mathbf{i} \right)+{{a}_{1}}{{b}_{2}}\left( \mathbf{i}\cdot \mathbf{j} \right)+{{b}_{1}}{{a}_{2}}\left( \mathbf{j}\cdot \mathbf{i} \right)+{{b}_{1}}{{b}_{2}}\left( \mathbf{j}\cdot \mathbf{j} \right) \\
& ={{a}_{1}}{{a}_{2}}\left( 1 \right)+{{a}_{1}}{{b}_{2}}\left( 0 \right)+{{b}_{1}}{{a}_{2}}\left( 0 \right)+{{b}_{1}}{{b}_{2}}\left( 1 \right) \\
& ={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}
\end{align}$
