How do you find a vector in the form <a,b,c> when only the angle between the two vectors?

Question

I have the angle between two 3D vectors $\beta$ and $\alpha$. Both their magnitude is 1. Can I calculate the vector $\beta$ and $\alpha$ from this angle and magnitude?

If anyone want to know how to calculate the angle between two vectors is simply $\theta = \arccos(\frac{\beta\cdot\alpha}{\left|\beta \right|\left|\alpha \right|})$

$\left|\beta \right|\left|\alpha \right| = 1$ for me, so the formula is simplified to $\theta = \arccos(\beta\cdot\alpha)$

Here is an example where an angle of 132 degrees is given. Example: NOT TO SCALE!!

Answer 1

You can take $\alpha$ to be the vector

$ \alpha = ( \sin t \cos s , \sin t \sin s , \cos t ) $

Then vector $\beta$ makes an anlge of $\theta$ with $\alpha$, therefore, define

the two unit vectors

$u_1 = (\cos t \cos s, \cos t \sin s , - \sin t ) $

$u_2 = (-\sin s , \cos s , 0 ) $

Since $ \alpha \cdot \beta = \cos \theta $ , then

$ \beta = \cos \theta \ \alpha + \cos r \sin \theta \ u_1 + \sin r \sin \theta \ u_2 $

And thus $\alpha $ and $\beta$ are determined by the arbitrary choice of $t, s$ and $r$.