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Angles

# Plane-Plane Angle

Two distinct planes in three-space either are parallel or intersecting in a line. If they intersect, you can determine the angle $(0\;\le\;\theta\;\le\;\frac{\pi}{2})\;$ between them from the angle between their normal vectors.

Specifically, if vectors n1=(A1,B1,C1) and n2=(A2,B2,C2) are normal to two intersecting planes,

$\pi_1\;\equiv\;A_1x+B_1y+C_1z+D_1=0\;$ and $\pi_2\;\equiv\;A_2x+B_2y+C_2z+D_2=0\;$

the angle θ between the normal vectors is equal to the angle between the two planes and is given by:

 That is:

Consequently, two planes with normal vectors n1 and n2 are

1. perpendicular if n1·n2=0
2. parallel if n2=kn1, for some nonzero scalar k
 Perpendicular planes Parallel planes

Determine the angle between the planes

$\fs2\;\pi_1\;\equiv\;2x-y+z-1=0\;and\;\pi_2\;\equiv\;x+z+3\;=\;0$

Solution:

n1=(2,-1,1) and n2=(1,0,1)

θ=30º