Question

Angle between two diagonals of a cube is :

(a) 30°

(b) 45°

(c) cos^-1(1/√3)

(d) cos^-1(1/3)

Answer 1

Answer is (d)

Let the adjacent cores of cube of side ‘a’ are OA, OB, OR to be taken as coordinate axis. 

Then the coordinates of the vertices of cube are following :

O(0, 0, 0), A(a, 0, 0), B(0, a, 0), R(0, 0, a), D(a, a, 0), K(a, 0, a), L(0, a, a,), P(a, a, a)

direction-cosines of diagonal Op are (a - 0, a - 0, a - 0) or portional to (a, a, a)

∴ Direction cosines of OP are

Similarly the direction cosines of AL, BK and RD are

Hence, the angle between two diagonals is cos^-1(1/3) an acute angle.