Project vector onto another vector a maximum angle $\theta$ I am looking for a very quick formula that would allow me to project/rotate a 3D unit vector $\hat{v}$ in the direction of another unit vector $\hat{v}_2$ some maximum angle $\theta$. I cannot do matrix operations, and performance is key since I have to perform this hundreds of billions of times. How would this behave if the angle chosen is larger than the angle between the two vectors? The other thing, though not crucial, is that it doesn't suffer from "gimbal lock", though if this makes it expensive then I can forgo it. Any help is appreciated. Thanks!

To rephrase your question: you have two unit length vectors $\hat v$ and $\hat v_2$ and an angle $\theta$ and you want a rotation by an angle of $\theta$ which fixes the plane spanned by $\hat v$ and $\hat v_2$ and with the angle oriented from $\hat v$ towards $\hat v_2$.

The plane spanned by $\hat v$ and $\hat v_2$ is perpendicular to the cross product of these two vectors. The cross product of that normal vector and $\hat v$ will result in a vector which lies not only in the plane but is also perpendicular to $\hat v$. You can use these two to establish a local coordinate system.

\begin{align*} v_3 &= \hat v\times\hat v_2 \\\ v_4 &= \hat v\times\hat v_3 \\\ \hat v_4 &= \frac{v_4}{\lVert v_4\rVert} \\\ \hat v_5 &= \cos\theta\,\hat v\pm\sin\theta\,\hat v_4 \end{align*}

Figuring out the correct sign is prone to errors, but you can simply plug in one example (e.g. $\hat v=(1,0,0)^T,\hat v_2=(0,1,0)^T$) and you will see that the negative sign is the correct one.