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Question

The angle between any two diagonals of a cube is:


A

cos1(12)

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B

cos1(13)

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C

cos1(14)

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D

π2

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Solution

The correct option is B

cos1(13)


Take one corner of the cube, O as the origin and OA, OB, OC - the three edges through it, as the axes.
Let, OA = OB = OC = a
The co-ordinates of the various corners are shown in the figure.



The four diagonals are AL, BM, CN and OP
The DRs of the diagonal AL are 0 - a, a - 0, a - 0,
i.e., -a,a,a
The DRs of diagonal OP are a - 0, a - 0, a - 0 i.e., a, a, a
If θ is the angle between them, then

cos θ=|(a)(a)+(a)(a)+(a)(a)|a2+a2+a2a2+a2+a2cos θ=a23a2=13θ=cos1(13)


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