3D coordinates and Angles

Question

A million years ago, an alien species built a vertical tower on a horizontal plane. 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), (1,1,0), and (0,3,3). By what angle does the tower now deviate from the vertical?

If you could explain the steps that you take to get to your answer, it would be appreciated!

Answer 1

To get you going:

(a) What was the equation of the original "horizontal" plane?

(b) What was the unit normal vector to that plane (pointing vertically upward)?

(c) What is the equation for the new tilted plane that is the base for the tower?

[Solve system of equations {ax+by+cz=d} for a,b,c,d with the three different sets of values for (x,y,z). The extra variable doesn't matter; think about that.]

(d) What is the unit normal vector to this new plane?

(e) What is the angle between the old normal and the new normal along the tower?