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Consider the two skew lines L1: r = (1, 2, 3) + s(2, 1, -1) and L2: r…
Question
Consider the two skew lines L1: r = (1, 2, 3) + s(2, 1, -1) and L2: r = (2, -1, 1) + u(1, 2, 1). The square of the shortest distance between them equals m/n in lowest terms, where m and n are positive integers. Find m + n.
✓ Verified answer: 7checked by our engine — not a guess
Step-by-step solution
The shortest distance between skew lines is |(a2 - a1)·(d1 × d2)| / |d1 × d2|.
d1 × d2 = |i j k; 2 1 -1; 1 2 1| = (1·1 - (-1)·2, (-1)·1 - 2·1, 2·2 - 1·1) = (3, -3, 3).
a2 - a1 = (1, -3, -2). Dot with (3, -3, 3): 3 + 9 - 6 = 6.
|d1 × d2|^2 = 9 + 9 + 9 = 27.
Distance^2 = 6^2 / 27 = 36/27 = 4/3.
So m/n = 4/3, m + n = 7.
Final answer7
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