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Why do the face diagonals have different angles?

  1. Jun 29, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the angle of the face diagonals of a (unit) cube.
    Screenshot - 06292015 - 07:59:34 AM.png Screenshot - 06292015 - 07:59:45 AM.png
    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).

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

    3. 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?
     
  2. jcsd
  3. Jun 29, 2015 #2

    Fredrik

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    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.
     
  4. Jun 29, 2015 #3
    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?
     
  5. Jun 29, 2015 #4

    Ray Vickson

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    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.
     
  6. Jun 29, 2015 #5
    Oh so that's what face diagonals mean. :blushing: Thank you.
     
  7. Jun 29, 2015 #6

    Fredrik

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