Let \(\mathbf{a}\) be a vector in \(\mathbb{R}^{n}\). Then, the dot product of the vector has the following property:
$$\mathbf{a}\cdot \mathbf{a}=|\mathbf{a}|^{2}$$
Proof
\begin{eqnarray*}&&\mathbf{a}\cdot \mathbf{a}=a_{1}a_{1}+a_{2}a_{2}+ \cdots +a_{n}a_{n}\\&&\ \ \ \ \ \ \ =a_{1}^{2}+a_{2}^{2}+ \cdots +a_{n}^{2}\\&&\ \ \ \ \ \ \ =|\mathbf{a}|^{2}\end{eqnarray*}