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Consider 12 face diagonal of a cubical block . How many pairs of them are skew lines .

I thought about it a lot . But don't get any idea . Can anybody help me in this .

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    If, by symmetry, you consider _one_ of the diagonals, which of the others are skew to that? Just count. – 2017-02-19
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    @HenningMakholm i think there would be 4 – 2017-02-19
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    If you start with one diagonal, there are four kinds of diagonal to consider. The one you first chose. Any diagonal parallel to the first. Any diagonal which meets the first. And the remainder which are skew to it. Each of the twelve diagonals fits into one of these categories - so you can check you have them all. Then remember you need to count each pair just once, rather than twice. – 2017-02-19
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    @user123733: You're missing one. Have you double checked the status of the two diagonals of the opposite face? – 2017-02-19
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    Diagonal on opposite face is parrallel not skew . And on opposite face there are two diagonals – 2017-02-19

1 Answers 1

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Draw a picture and note that each diagonal has 5 skew lines to make pairs with.

So, 12 * 5 = 60. However, we have counted each pair twice. So, divide by two to get your answer: 30

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    Each diagonal would have 4 skew lines – 2017-02-20
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    Each diagonal on a face contains one diagonal on each of the other faces to which the face diagonal is skew. This means that there are 5 pairs of face diagonals. – 2017-02-20
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    A simpler explanation is that there are five skew lines for each diagonal since there is one on each other face. – 2017-02-20
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    But the opposite face diagonal is parrallel not skew – 2017-02-20
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    If one diagonal goes from the point (0, 0, 0) to (1, 1, 1), it can be represented by the vector <1, 1, 1> the opposite diagonal would be the vector <1, 1, 0>. The angle between these two are approximately 35 degrees. So, they are definitely not perpendicular/intersecting lines. – 2017-02-20
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    The diagonal (0,0,0) to (1,1,1) is body diagonal – 2017-02-20