Two ropes are tied along the two diagonals of a cubic room. Ropes are inclined to each other at :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\cos^{-1}\left(\displaystyle\frac{1}{3}\right)\)
The ropes run along two diagonals of the cube, each joining a corner to the opposite corner through the inside of the room. Take one corner of the room as the origin and the three edges through it along the axes, with side a.
One diagonal joins (0, 0, 0) to (a, a, a); another joins (a, 0, 0) to (0, a, a). Their direction ratios are
\((a, a, a)\ \text{or}\ (1, 1, 1)\)
\((-a, a, a)\ \text{or}\ (-1, 1, 1)\)
The angle between two lines with direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is given by
\( \displaystyle \cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}} \)
\( \displaystyle = \frac{|-1 + 1 + 1|}{\sqrt{3}\,\sqrt{3}} = \frac{1}{3} \)
So \(\theta = \cos^{-1}\left(\frac{1}{3}\right)\). Any two of the four diagonals give the same value, because the cube looks the same from every corner.
Why the other options are incorrect
Option (1) \(45^{\circ}\) — Incorrect. \(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\), not \(\frac{1}{3}\); 45° is the angle between an edge and a diagonal of a face, which lies on one wall.
Option (2) \(30^{\circ}\) — Incorrect. \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), not \(\frac{1}{3}\).
Option (3) \(\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\) — Incorrect. This is the angle between a diagonal of the cube and an edge: with (1, 1, 1) and (1, 0, 0), \(\cos\theta = \frac{1}{\sqrt{3}\times 1}\).