Find Angle Between Vectors C & D Using Alpha, Beta & Theta

[meanyack]

Homework Statement

Please find the attachment for vectors. If I know $$\alpha$$, $$\beta$$ & $$\theta$$ (colored angles) how can I find the angle between vectors C and D in terms of $$\alpha$$, $$\beta$$ & $$\theta$$?

Homework Equations

Cross product & dot product

The Attempt at a Solution

 

[johnsonc007]

The cross product is defined by u X v = |u||v|sin(angle). Solve for the angle.

 

[meanyack]

At first glance, it seems easy. But when you try to find the angle between vector C and plane consisting A and B, then it seems complicated. So is there any way to calculate it easily?