Why do the face diagonals have different angles?

[raddian]

Homework Statement

Find the angle of the face diagonals of a (unit) cube.

I agree with this solution, but I have a problem with another face diagonal: the face diagonal from the angle (0,0,1),(0,0,0), and (1,1,1).

Homework Equations

dot product
cos-1(a.b/ (|a||b|)

The Attempt at a Solution

From the solution, we have an angle given from the points (1,0,1), (0,0,0), and (0,1,1).
Using the def. of dot product, if A is the vector (0,0,0) to (1,0,1) and B is the vector (0,0,0) to (0,1,1)
cos-1(A.B/(|A||B|)), where A.B = 1, |A| = |B| = sqrt(2).
Thus cos-1(1/ [ (sqrt(2)sqrt(2) ]) = 60 deg. Ok.

the face diagonal from the angle (0,0,1),(0,0,0), and (1,1,1):
A is the vector (0,0,0) to (0,0,1) and B is the vector (0,0,0) to (1,1,1)
cos-1(A.B/(|A||B|)), where A.B = 1, |A| = 1, |B| = sqrt(3).
thus cos-1(1/ [ sqrt(3) ]) != 60 deg.

Why is this happening?

 

[Fredrik]

The line from (0,0,0) to (0,0,1) is an edge, not a face diagonal. The line from (0,0,0) to (1,1,1) is an "inside" diagonal, not a face diagonal.

 

[raddian]

Isn't the line from (0,0,1) to (1,1,1) a face diagonal?

The line from (0,0,0) to (0,0,1) is an edge, not a face diagonal. The line from (0,0,0) to (1,1,1) is an "inside" diagonal, not a face diagonal.

The angle made with the edge and the inside diagonal is the angle of the face diagonal, assuming the face diagonal is (0,0,1) to (1,1,1), right?

 

[Ray Vickson]

Isn't the line from (0,0,1) to (1,1,1) a face diagonal?

The angle made with the edge and the inside diagonal is the angle of the face diagonal, assuming the face diagonal is (0,0,1) to (1,1,1), right?

Wrong: look again at the diagram. You want the angle between *two* face-diagonals originating at the same base-point and lying in two of the faces that meet at that point. So, the vector from (0,0,1) to (1,1,1) is a face diagonal. but the vector from (0,0,1) to (0,0,0) is not.

 

[raddian]

You want the angle between *two* face-diagonals originating at the same base-point and lying in two of the faces that meet at that point.

Oh so that's what face diagonals mean. Thank you.

 

[Fredrik]

Isn't the line from (0,0,1) to (1,1,1) a face diagonal?

Yes it is. But you're looking for the angle between the two lines I mentioned, and they are not face diagonals.