How is $x^2$ related to $x^Tx$

Question

If $x$ is a column vector how is $x^2$ related to $x^\top x$?

Where $x^\top$ is the transpose of $x$.

Answer 1

Let $\|x\|$ be the norm-2 (euclidean norm) of $x$. Then:

$$x^\top x = \|x\|^2.$$

Someone has the bad habit to assume $x^2 = \|x\|^2$, but it is not rigorously correct.

Answer 2

Let $ x \in \mathbb{R}^n $ and $ \left \| \cdot \right \| $ be the euclidean norm,
$ \left \| x \right \|^2 = < x, x > = ^tx \cdot x $