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Question
The angle between the two diagonals of a cube is
Options
(a) 30°
(b) 45°
(c) \[\cos^{- 1} \left( \frac{1}{\sqrt{3}} \right)\]
(d) \[\cos^{- 1} \left( \frac{1}{3} \right)\]
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Solution
\[\cos^{- 1} \left( \frac{1}{3} \right)\]
\[\text { Let a be the length of an edge of the cube and let one corner be at the origin as shown in the figure . Clearly, OP, AR, BS, and CQ are the diagonals of the cube } . \]
\[\text{ Consider the diagonals OP and AR } . \]
\[\text{ Direction ratios of OP and AR are proportional to a - 0, a - 0, a - 0 and 0 - a, a - 0, a - 0, i . e . a, a, a and - a, a, a, respectively } . \]
\[\text { Let } \theta \text{ be the angle between OP and AR . Then,} \]
\[\cos \theta = \frac{a \times - a + a \times a + a \times a}{\sqrt{a^2 + a^2 + a^2}\sqrt{\left( - a \right)^2 + a^2 + a^2}}\]
\[ \Rightarrow \cos \theta = \frac{- a^2 + a^2 + a^2}{\sqrt{3 a^2}\sqrt{3 a^2}}\]
\[ \Rightarrow \cos \theta = \frac{1}{3} \]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{1}{3} \right) \]
\[\text{ Similarly, the angles between other pairs of the diagonals are equal to } \cos^{- 1} \left( \frac{1}{3} \right) \text{ as the angle between any two diagonals of a cube is } \cos^{- 1} \left( \frac{1}{3} \right) .\]
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