Creating a vector with the same direction, but different magnitude

Question

Let's say I have a vector $u$, whose magnitude (L2-norm) is $|u|$ and whose direction is $\frac{u}{|u|}$. I have another vector $v$, whose magnitude and direction are both different to that of $u$. So, $|v| \neq |u|$ and $\frac{v}{|v|} \neq \frac{u}{|u|}$.

What I want to do, is create a new vector $w$ whose magnitude is equal to that of $u$, but whose direction is equal to that of $v$. So this can be thought of as taking vector $v$, and changing its magnitude such that the magnitude is equal to $|u|$.

This seems quite trivial, but I cannot work it out! Any help please?

Answer 1

$$w = |u|\frac{v}{|v|}{}{}{}$$

Answer 2

As a general rule, given a nonzero element of a vector space, that is, a vector $v$, take any nonzero scalar $\alpha$ not equal to $1$, and the product $\alpha v$ will do the trick.

For your specific question, which is different (more detailed) than the title, just choose $\alpha=|u|/|v|$.

At least, unless there is a pathological example where this isn't true ( which at the moment eludes me). But it should be true in $L^2$, which is actually a Hilbert space.