Vectors and Trig and Angles?

Two geometry questions here:





1. Let A and B be two angles in 3d space. They share a point of origin, and one side. The planes containing these angles are perpendicular. What is the angle between the two outer sides of these angles, expressed in terms of A and B?





2. Let x+y+z=V be a vector in 3d space with its point at the origin. A is the angle between V and the x-axis, B is the angle between V and the y-axis, and C is the angle between V and the z-axis.


What is the value of C in terms of B and A?

Vectors and Trig and Angles?
These are fun, and you really should work them on your own. Both involve using dot products.





But, if you want them worked...and no guarantees these are right, it's 2 AM...


(First let it be defined that i, j, and k are the unit vectors on the X, Y, and Z axes, respectively.)


1. Let's assume these start from the origin, that the side they share is the Y axis, that A stretches onto the X axis, and that B stretches onto the Z axis. Then Va = j cos A + i sin A, and Vb = j cos B + k sin B. The easiest thing to do with these is take the dot product: Va . Vb = cos A cos B. However, we know the dot product is |Va| |Vb| cos theta. Since |Va| and |Vb| are both 1, we have cos theta = cos A cos B. Now we know theta (or we would, if I bothered to remember the trig identity you use here). Anyway, at this point it isn't hard to find theta.





2. Without loss of generality let V be a unit vector (or scale it until it is). Then V . i = cos A, V . j = cos B, and V . k = cos C. However, since V is a unit vector, (V . i)^2 + (V . j)^2 + (V . k)^2 = 1^2 = 1, or cos^2 A + cos^2 B + cos^2 C = 1. From here you can isolate cos C in terms of A and B as sqrt(1 - cos^2 A - cos^2 B), and hopefully there's yet another trig identity I don't remember that will simplify this.





Anyway, yeah, it's mainly converting between the algebraic and geometric definitions of the dot product. What you're given is angles, but what's easiest to manipulate is quantities. So that's why we ended up in both cases with arc-trig-function of algebra-combination of trig-functions.