# Creating a vector with the same direction, but different magnitude

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?

$$w = |u|\frac{v}{|v|}{}{}{}$$
• I kinda hate that $u$ and $v$ are the usual symbols for this sort of thing, they're hard to tell apart. – Dan Uznanski Jan 26 at 21:01
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.