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Let Π be the plane through the three points A = (1, 0, 2), B = (3,…
Question
Let Π be the plane through the three points A = (1, 0, 2), B = (3, -1, 1) and C = (2, 2, 4). The distance from the point Q = (5, 4, -3) to the plane Π equals d. The value of d^2 equals m/n in lowest terms with m, n positive integers. Find m + n.
✓ Verified answer: 83checked by our engine — not a guess
Step-by-step solution
A normal to the plane is n = (B - A) × (C - A). B - A = (2, -1, -1), C - A = (1, 2, 2).
n = ((-1)(2) - (-1)(2), (-1)(1) - (2)(2), (2)(2) - (-1)(1)) = (0, -5, 5).
Distance from Q to plane = |n·(Q - A)| / |n|. Q - A = (4, 4, -5).
n·(Q - A) = 0 - 20 - 25 = -45; |n| = sqrt(0 + 25 + 25) = sqrt(50).
d = 45/sqrt(50), so d^2 = 2025/50 = 81/2.
Then m + n = 81 + 2 = 83.
Final answer83
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