Question

A: The acute angle between the space diagonals (passing through opposite corners) of a cube.
B: The angle between the face diagonals of a cube.
C: The acute angle between a face diagonal and a space diagonal of a cube passing through the same corner of the cube.
Arrange A, B, C in ascending order

• A. A, B ,C
• B. B, C, A
• C. B, A, C
• D. C, A, B

Answer 1

The correct option is D C, A, B

Let us consider a unit cube with one of the vertices as the origin and the coordinate axes along the edges of the cube. We can find the angles between the given lines using dot product of vectors along the lines.
The angle between the two diagonals of a cube is ${\theta }_1={\cos }^{-1}\frac{1}{3}$
The angle between two diagonals of a face of cube is ${\theta }_2=\frac{\pi }{2}$
The angle between a diagonal of a cube and diagonal of face of a cube is ${\theta }_3={\cos }^{-1}\sqrt{\frac{2}{3}}$
Arranging in ascending order,
${\theta }_3,{\theta }_1,{\theta }_2$
Hence, option D is the correct answer.