# Solve for the third dihedral angle in an intersection of three planes

Three planes A, B and C intersect at point P. The dihedral angle between A and B is $\theta$ and the dihedral angle between B and C is $\psi$.

1. Solve for the dihedral angle between by A and C.

2. Planes A and B intersect plane C in two lines that form angle QPR. That is, point Q lies in both A and C, and point R lies in both B and C. Solve for the angle of QPR in terms of $\theta$ and $\psi$.

3. Does this fall under trigonometry? If not, what?

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You may want to tag this as differential geometry too. –  user3180 Nov 29 '10 at 8:58
No, differential geometry involves derivatives. This is pure vector geometry. –  Hans Lundmark Nov 29 '10 at 9:05

Question 1: The third angle is not uniquely determined by $\theta$ and $\psi$. If $a$, $b$ and $c$ are the unit normal vectors to the planes $A$, $B$ and $C$, then the problem is equivalent to determining $a \cdot c$ from $a \cdot b$ and $c \cdot b$, which cannot be done. (Think of $b$ being fixed; then giving values of $a \cdot b$ and $c \cdot b$ constrains $a$ and $c$ to lie on certain cones around $b$, but the positions of $a$ and $c$ on these cones are arbitrary.)