HMMT 十一月 2015 · 冲刺赛 · 第 28 题
HMMT November 2015 — Guts Round — Problem 28
题目详情
- [ 15 ] Find the shortest distance between the lines = = and = = 2 3 1 − 1 1 2
解析
- [ 15 ] Find the shortest distance between the lines = = and = = 2 3 1 − 1 1 2 Proposed by: Sam Korsky √ 5 3 Answer: 3 First we find the direction of a line perpendicular to both of these lines. By taking the cross product (2 , 3 , 1) × ( − 1 , 1 , 2) = (5 , − 5 , 5) we find that the plane x − y + z + 3 = 0 contains the first line and is parallel to the second. Now we take a point on the second line, say the point (3 , 0 , − 1) and find the √ 5 3 | 3 − 0+( − 1)+3 | 5 √ √ distance between this point and the plane. This comes out to = = . 2 2 2 1 +1 +1 3 3