Finding the direction vector magnitude from position and angle

I'm writing a piece of code for one astronomical problem, basically I want a fast way to check if the field of view of two meteor cameras is overlapping so I can proceed to feed the data into a meteor trajectory solver.

I was able to simplify my problem to this little graph: Problem

In summary: I have a vector $$\vec{b}$$ (representing the centre of the field of view of one camera at the height of 100 km) and a unit direction vector $$\hat{c}$$ pointing to the same point in another camera's field of view. I would like to find a vector $$\vec{a}$$ which is basically $$\vec{a} = \vec{b} + k \hat{c}$$ so that the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$\theta$$. Finding either $$\vec{a}$$ or the scalar $$k$$ would do the trick.

I'm ashamed that my knowledge of trigonometry is so rusty that I can't solve what seems to be a simple problem! I tried attacking it using the law of cosines, but to no avail.

All help would be much appreciated!

Cheers

the dot product $${{{\bf \hat c} \cdot {\bf \vec b}} \over {\left| {{\bf \hat c}} \right|\left| {{\bf \vec b}} \right|}} = \cos \left( {\pi - \alpha } \right)$$ will give you the angle $$\alpha$$: the one opposite to $$\bf \vec a$$.
After that apply the "law of sines" to find the length of $$\bf \vec c$$.