Question: Take a look at the image bellow:

![enter image description here](https://i.stack.imgur.com/M6Yl8.png)

How can I find the point P, without knowing the vector B (or its end point)?
Known is:

* The vector A (start point, end point and its length)
* Angle theta

Answer: 
You know the length/magnitude of vector $OP$ where $O$ is the vertex. You also know the vector $OA$. Find the unit vector in the direction of $OA$. Let us call it $e = (\cos\alpha , \sin\alpha)$. Now the direction of vector $OP$ is also known : $a = (\cos(\alpha - \theta),\sin(\alpha-\theta)) $ if $\theta $ is defined counter-clockwise.
Since you know $A$, you can find the angle $A$ makes with the horizontal axis. Call this angle $\varphi$.

Then $\varphi-\theta$ is the angle that $B$ makes with the $x$-axis. You can now find the $x$ and $y$ coordinates using right-triangle trig. (The hypotenuse of the triangle of interest is $\|A\|\cos\theta$.)

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Another approach (that's more work than it's worth, but still an option) is to use the rotation matrix to make $A$ parallel to $B$. Then compute the projection of $A$ onto this rotated vector $A$. (This would be how I'd do it if I had a computer algebra system and only needed to solve the problem once.)