Jamarcus Lindsey

2022-10-01

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, $|v|\ne |u|$ and $\frac{v}{|v|}\ne \frac{u}{|u|}$
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 & Explanation

Samantha Braun

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

Austin Rangel

As a general rule, given a nonzero element of a vector space, that is, a vector v, take any nonzero scalar α not equal to 1, and the product αv 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.

Do you have a similar question?

Recalculate according to your conditions!