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Maths Exercices

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 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,

and

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



Solution:

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

θ=30º