Find $y$ such that $\langle v, y \rangle = \|v\|$

Question

Let $v \in \mathbb{R^n}$. Show which vector $y \in \mathbb{R^n}$ satisfies:

$\langle v, y \rangle = \|v\| := \sqrt{ \langle v, v \rangle}$.

Where $\langle \cdot, \cdot \rangle$ is the usual inner product.

Any ideas? Thanks!

Answer 1

For each $x \not\perp v$ we have $y=\frac{||v||x}{<v,x>}$ satisfies the condition. Now they are the only ones with this property. So, for each direction not ortoghonal to the line spanned by v exists one and only one vector with that property.