The angle between any two diagonals of a cube is:
cos−1(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+a2√a2+a2+a2cos θ=a23a2=13⇒θ=cos−1(13)