HMMT 十一月 2015 · 冲刺赛 · 第 28 题

英文解析

28. [ 15 ] Find the shortest distance between the lines = = and = =
    2 3 1 − 1 1 2
    Proposed by: Sam Korsky
    √
    5 3
    Answer:
    First we find the direction of a line perpendicular to both of these lines. By taking the cross product 3
    (2 , 3 , 1) × ( − 1 , 1 , 2) = (5 , − 5 , 5) we find that the plane x − y + z + 3 = 0 contains the first line and isparallel 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