If each side of a cube is x, then the angle between the diagonals of the cube is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\cos^{-1}\left(\displaystyle\frac{1}{3}\right)\)
Place the cube with one corner at the origin and edges of length x along the axes. Two of its diagonals are
\(\text{from } (0, 0, 0) \text{ to } (x, x, x):\ \text{direction ratios } x, x, x\)
\(\text{from } (x, 0, 0) \text{ to } (0, x, x):\ \text{direction ratios } -x, x, x\)
The angle between two lines with direction ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) is given by
\( \displaystyle \cos\theta = \left|\frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}\right| \)
\( \displaystyle \cos\theta = \left|\frac{-x^2 + x^2 + x^2}{\sqrt{3}\,x \cdot \sqrt{3}\,x}\right| = \frac{x^2}{3x^2} = \frac{1}{3} \)
\( \displaystyle \theta = \cos^{-1}\left(\frac{1}{3}\right) \)
By symmetry every pair of diagonals gives the same value. Two intersecting lines make two angles, \(\theta\) and \(\pi - \theta\); the angle between the lines is taken as the acute angle, which is why the absolute value is used.
Why the other options are incorrect
Option (1) \(\cos^{-1}\left(\displaystyle\frac{1}{\sqrt{3}}\right)\) — Incorrect. This is the angle between a diagonal (1, 1, 1) and an edge (1, 0, 0): \(\cos\theta = \frac{1}{\sqrt{3}\cdot 1}\).
Option (3) \(\cos^{-1}\left(-\displaystyle\frac{1}{\sqrt{3}}\right)\) — Incorrect. This is the obtuse angle between a diagonal and an edge, not between two diagonals.
Option (4) \(\cos^{-1}\left(-\displaystyle\frac{1}{3}\right)\) — Incorrect. This is \(\pi - \theta\), the obtuse angle made by the same two diagonals; the angle between two lines is the acute one.