A million years ago, an alien species built a vertical tower on a horizontal plane.

Question

When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0,0,0) , (1,2,0) and (0,3,1) . By what angle does the tower now deviate from the vertical?

Answer 1

a normal to the xy plane is (0,0,1)=q

a normal to the tilted plane is cross product of

(2,1,0) and (0,3,1) which equals (2,1,3)=r

the scalar product of the two normals=3

also the scalar product of the two normals=|q||r|cos(angle)

angle is 36.7 degrees.