[Math4u] Given two 3-d vectors, determine angle

Hi everyone. I can't seem to figure this out. Can someone give me
some guidance?

Objective: Given two normal vectors, determine the angle between them
from 0 to 360 degrees. The 0 to 360 is IMPORTANT because I need to
know which quadrant it is in.

I have attempted to use the "Dot Product". However, this only returns
a value from 0 to 180. Also, due to the nature of cosine, there are
multiple vectors that can give the same angle. [ie (1,0,0) dot
(1,1,0) and (1,0,0) dot (1,-1,0)]. Both equate to 45 degrees.
However, I can't figure out how to determine if it is 45 degrees or 315.

I was thinking I could check to the X & Y values to determine their
sign.

X | Y | QUADRANT
----------------
+ | + | I (0 to 90)
- | + | II (90 to 180)
- | - | III (180 to 270)
+ | - | IV (270 to 360)

But, what do I do with the Z!?? Since the dot product for 3d space is
x1*x2 + y1*y2 + z1*z2, I cannot ignore the Z values. But, I don't
know how to use it to determine the quadrant/correct angle.

Any guidance would be appreciated! Thanks!

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