已知A(M,1+M,2+M),B(1-M,3-2M,3M)是空间两个动点,求AB向量模的最小值

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已知A(M,1+M,2+M),B(1-M,3-2M,3M)是空间两个动点,求AB向量模的最小值
已知A(M,1+M,2+M),B(1-M,3-2M,3M)是空间两个动点,求AB向量模的最小值

已知A(M,1+M,2+M),B(1-M,3-2M,3M)是空间两个动点,求AB向量模的最小值
AB = OB-OA
= (1-2m,2-3m,-2+2m)
|AB|^2 = (1-2m)^2 + (2-3m)^2+ (-2+2m)^2
(|AB|^2)' = -4(1-2m)-6(2-3m)+4(-2+2m)
=-24+34m =0
=>m = 12/17
(|AB|^2)'' = 34 > 0 ( min)
min(|AB|)
= [ (1-2(12/17))^2 + (2-3(12/17))^2+ (-2+2(12/17))^2] ^(1/2)
= (49+4+100)^(1/2)/17
=(3√17) / 17