Maths · Direction ratios and direction cosines and the angle between two intersecting lines
If a ray makes angles and with the four diagonals of a cube and Arrange in desce
If a ray makes angles \( \alpha, \beta, \gamma \) and \( \delta \) with the four diagonals of a cube and \( \mathbf{A}: \cos ^{2} \boldsymbol{\alpha}+\cos ^{2} \boldsymbol{\beta}+\cos ^{2} \boldsymbol{\gamma}+\cos ^{2} \boldsymbol{\delta} \) \( \mathbf{B}: \sin ^{2} \boldsymbol{\alpha}+\sin ^{2} \boldsymbol{\beta}+\sin ^{2} \boldsymbol{\gamma}+\sin ^{2} \boldsymbol{\delta} \) \( \mathbf{C}: \cos 2 \boldsymbol{\alpha}+\cos 2 \boldsymbol{\beta}+\cos 2 \gamma+\cos 2 \boldsymbol{\delta} \) Arrange \( A, B, C \) in descending order
- A. \( B, A, C \)
- B. \( A, B, C \)
- C. \( C, A, B \)
- D. \( B, C, A \)
Step-by-step solution
Let the ray have direction cosines (l,m,n). The cosines of angles with the four diagonals are (l+m+n)/√3, (l-m-n)/√3, (-l+m-n)/√3, (-l-m+n)/√3. Their squares sum to 4/3 (since l²+m²+n²=1). Thus A = 4/3. B = sum sin² = 4 - sum cos² = 8/3. C = sum cos2θ = 2 sum cos²θ - 4 = -4/3. Descending order: B (8/3) > A (4/3) > C (-4/3).
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