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 how you got to your answer, it would be appreciated!

Answer 1

It is done in a way very similar to your other question.

Here are the steps.

1) Find equation of the plane containing those three given points.

2) The gradient is the vector of the tower.

3) The dot product between the vector of the tower and the vertical vector gives the cosine of the angle.

1)

In general, a plane has the equation

ax + by + cz = d

We find the coefficients a, b, c, d by substituting in the coordinates of the three points.

That gives us three equations and four unknowns.

But it is not really four unknowns because there are many equations for the same plane,

all differing only by a constant factor -- so we have only three degrees of freedom.

a×0 + b×0 + c×0 = d

a×1 + b×1 + c×0 = d

a×0 + b×3 + c×3 = d

That simplifies into

0 = d

a + b = 0

b + c = 0

One possible solution is

a = 1, b = -1, c = 1, d = 0

All other solutions are constant multiples of this one.

So an equation of the plane is

x - y + z = 0

2)

The gradient of the function f(x, y, z) = x - y + z is

∇f = (1, -1, 1)

That is a vector of the tower.

(Every other vector of the tower is a constant multiple of this one.)

3)

In general, the dot product u.v of two vectors u, v has the property

cos(α) = u.v/(|u||v|)

where α is the angle between u and v.

A vector of the vertical is v = (0, 0, 1).

The dot product

∇f.v = 1×0 + (-1)×0 + 1×1 = 1

|∇f| = √(1² + (-1)² + 1²) = √3

|(0, 0, 1)| = 1

Thus

cos(α) = 1/√3

α ≈ 54.7°