Answer
.
Work Step by Step
Consider the given identity
$\mathbf{u}\cdot \left( \mathbf{v}+\mathbf{w} \right)=\mathbf{u}\cdot \mathbf{v}+\mathbf{u}\cdot \mathbf{w}$ ...… (1)
For the given vectors u, v, and w,
$\mathbf{u}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$
$\mathbf{v}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$
$\mathbf{w}={{a}_{3}}\mathbf{i}+{{b}_{3}}\mathbf{j}$
Now, take the left side of equation (1) to get
$\mathbf{u}\cdot \left( \mathbf{v}+\mathbf{w} \right)$
The left side of the given identity represents the dot product of two vectors $\mathbf{u}$ and $\left( \mathbf{v}+\mathbf{w} \right)$.
Consider, $\mathbf{h}=\mathbf{v}+\mathbf{w}$
$\begin{align}
& \mathbf{h}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{i}+{{b}_{3}}\mathbf{j} \\
& \mathbf{h}=\left( {{a}_{2}}+{{a}_{3}} \right)\mathbf{i}+\left( {{b}_{2}}+{{b}_{3}} \right)\mathbf{j} \\
\end{align}$
Now,
$\mathbf{u}\cdot \left( \mathbf{v}+\mathbf{w} \right)=\mathbf{u}\cdot \mathbf{h}$ …… (2)
Substituting the value of $h$ in (2), we get
$\mathbf{u}\cdot \mathbf{h}={{a}_{1}}\left( {{a}_{2}}+{{a}_{3}} \right)\mathbf{i}+{{b}_{1}}\left( {{b}_{2}}+{{b}_{3}} \right)\mathbf{j}$
$\mathbf{u}\cdot \left( \mathbf{v}+\mathbf{w} \right)={{a}_{1}}\left( {{a}_{2}}+{{a}_{3}} \right)\mathbf{i}+{{b}_{1}}\left( {{b}_{2}}+{{b}_{3}} \right)\mathbf{j}$ …… (3)
Consider the right side of (1).
The right side of the given identity represents addition of dot products of $\mathbf{u}\,\mathbf{v}$ and $\mathbf{u}\,\mathbf{w}$. Therefore,
$\mathbf{u}\cdot \mathbf{v}={{a}_{1}}{{a}_{2}}\mathbf{i}+{{b}_{1}}{{b}_{2}}\mathbf{j}$
$\mathbf{u}\cdot \mathbf{w}={{a}_{1}}{{a}_{3}}\mathbf{i}+{{b}_{1}}{{b}_{3}}\mathbf{j}$
$\begin{align}
& \mathbf{u}\cdot \mathbf{v}+\mathbf{u}\cdot \mathbf{w}={{a}_{1}}{{a}_{2}}\mathbf{i}+{{b}_{1}}{{b}_{2}}\mathbf{j}+{{a}_{1}}{{a}_{3}}\mathbf{i}+{{b}_{1}}{{b}_{3}}\mathbf{j} \\
& ={{a}_{1}}{{a}_{2}}\mathbf{i}+{{a}_{1}}{{a}_{3}}\mathbf{i}+{{b}_{1}}{{b}_{2}}\mathbf{j}+{{b}_{1}}{{b}_{3}}\mathbf{j}
\end{align}$
Solving above equation gives
$\mathbf{u}\cdot \mathbf{v}+\mathbf{u}\cdot \mathbf{w}={{a}_{1}}\left( {{a}_{2}}+{{a}_{3}} \right)\mathbf{i}+{{b}_{1}}\left( {{b}_{2}}+{{b}_{3}} \right)\mathbf{j}$ …… (4)
Equations (3) and (4) are identical.
So,
Left side of equation (1) = Right side of equation (1)
Hence, $\mathbf{u}\cdot \left( \mathbf{v}+\mathbf{w} \right)=\mathbf{u}\cdot \mathbf{v}+\mathbf{u}\cdot \mathbf{w}$.
