How can I compute $\max\{v^Tx :\, v \in \mathbb{R}^n \textrm{ and } \|v\| = 1\}$?

Question

If $x \in \mathbb{R}^n$, how can I compute this?

$$\max\{v^Tx :\, v \in \mathbb{R}^n \textrm{ and } \|v\| = 1\}$$

here, $\|\cdot\|$ is euclidean norm.

Answer 1

The maximum value is $||x||$ because you can observe that

$v^tx=||v||||x||\cos(\theta(v,x))=||x||\cos(\theta(v,x))$

Thus the maximum value there is only for $v=\lambda x$ but $||v||=1$ so

$$|\lambda| =\frac{1}{||x||}$$

This means that

$$\sup_v v^tx=\frac{x^t}{||x||}x=\frac{||x||^2}{||x||}=||x||$$