Answer
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Work Step by Step
Consider the given identity
$\mathbf{u}\cdot \mathbf{v}=~~\mathbf{v}\cdot \mathbf{u}$ …… (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).
The left-hand side of the given identity represents the dot product of vectors $\mathbf{u}\ \text{ and }\ \mathbf{v}$. It is calculated as follows:
$\mathbf{u}\cdot \mathbf{v}={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}$ …… (2)
Now, take the right side of equation (1).
The right side of the given identity represents the dot product of $\mathbf{v}\ \text{ and }\ \mathbf{u}$. It is calculated as follows:
$\mathbf{v}\cdot \mathbf{u}={{a}_{2}}{{a}_{1}}+{{b}_{2}}{{b}_{1}}$ …… (3)
Equations (2) and (3) are identical.
So,
Left side of equation (1) = Right side ofequation (1)
Hence, $\mathbf{u}\cdot \mathbf{v}=~~\mathbf{v}\cdot \mathbf{u}$.
