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Let L be the line of intersection of the planes x + 2y - z = 1 and 2x…
Question
Let L be the line of intersection of the planes x + 2y - z = 1 and 2x - y + z = 3. Let θ be the angle between L and the plane 2x + 2y + z = 5. Compute floor(1000·sin^2 θ).
✓ Verified answer: 257checked by our engine — not a guess
Step-by-step solution
The direction of L is d = n1 × n2 where n1 = (1, 2, -1), n2 = (2, -1, 1).
d = (2·1 - (-1)(-1), (-1)(2) - (1)(1), (1)(-1) - (2)(2)) = (1, -3, -5).
For a line making angle θ with a plane of normal n3 = (2, 2, 1): sin θ = |d·n3| / (|d||n3|).
d·n3 = 2 - 6 - 5 = -9; |d|^2 = 1 + 9 + 25 = 35; |n3|^2 = 4 + 4 + 1 = 9.
sin^2 θ = 81 / (35·9) = 81/315 = 9/35 ≈ 0.257142...
1000·sin^2 θ ≈ 257.14, so floor = 257.
Final answer257
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