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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?

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

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

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