# Why do the face diagonals have different angles?

## 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?

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Fredrik
Staff Emeritus
Gold Member
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.

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
Homework Helper
Dearly Missed
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.

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
Staff Emeritus