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A straight line with direction ratios (1, 2, 2) intersects the planes 3x - 2y + 6z + 12 = 0 and 3x - 2y + 6z - 2 = 0 at points P and Q respectively. If the length of segment PQ is lambda, find the value of [lambda], where [.] denotes the greatest integer (floor) function.
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Correct answer: 3
Solution
The two planes are parallel with perpendicular separation 2. The line's unit direction vector has a component 11/21 along the common normal, so PQ = 2/(11/21) = 42/11 ≈ 3.818. Thus [lambda] = [3.818] = 3.
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