Answer
See the explanation below.
Work Step by Step
Consider the given identity
$\left( c\mathbf{u} \right)\cdot \mathbf{v}=c\left( \mathbf{u}\cdot \mathbf{v} \right)$ …… (1)
For the given vectors
$\mathbf{u}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$
$\mathbf{v}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$
Now, take the left side of equation (1).
$c\mathbf{u}=c{{a}_{1}}\mathbf{i}+c{{b}_{1}}\mathbf{j}$
$\mathbf{v}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$
The left side of the given identity represents the dot product of two vectors $c\mathbf{u}\ \text{ and }\ \mathbf{v}$ and it is calculated as
$\left( c\mathbf{u} \right)\cdot \mathbf{v}=c{{a}_{1}}{{a}_{2}}+c{{b}_{1}}{{b}_{2}}$ …… (2)
Now, take the right side of equation (1).
The right side of the given identity represents the dot product of two vectors $c\mathbf{u}\ \text{ and }\ \mathbf{v}$ multiplied by $c$ and it is calculated as
Dot product of $\mathbf{u}\ \text{ and }\ \mathbf{v}$ is
$\mathbf{u}\cdot \mathbf{v}={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}$ …… (3)
Multiply equation (3) by $c$ to get
$c(\mathbf{u}\cdot \mathbf{v})=c{{a}_{1}}{{a}_{2}}+c{{b}_{1}}{{b}_{2}}$ …… (4)
From equations (2) and (4), we get: the left side of equation (1) is equal to the right side of equation (1).
Thus,
$\left( c\mathbf{u} \right)\cdot \mathbf{v}=c\left( \mathbf{u}\cdot \mathbf{v} \right)$
