(1)在平面ABCD内过点B作AC的平行线BE, ∵AC∥A1C1,AC∥BE, ∴BE∥A1C1, ∴面A1BC1与面ABCD的交线l与BE重合, 即直线BE就是所求的直线l. ∵BE∥A1C1, l与BE重合, ∴l∥A1C1. (2)证明:连接B1D1, ∵A1B1C1D1是正方形, ∴A1C1⊥B1D1, ∵A1C1⊥DD1, ∴A1C1⊥面DBB1D1, ∴A1C1⊥B1D. 同理A1B⊥面ADC1B1, ∴A1B⊥B1D, ∵A1C1∩A1B=A1, ∴B1D⊥面A1BC1. (3)∵AC∥A1C1,且AC在面A1BC1外,A1C1⊂面A1BC1, ∴AC∥面A1BC1, ∴直线AC到面A1BC1的距离即为点A到面A1BC1的距离,记为h, 在三棱锥中A-A1BC1中, VA_A1BC1=VC1-ABA1, ∵正方体A1B1C1D1-ABCD棱长为a, ∴VA-A1BC1=•S△A1BC1•h=××(a)2×h×sin60°=h, VC1-ABA1=•S△ABA1•A1C1=••a•a•a=a3, ∵VA_A1BC1=VC1-ABA1, ∴h=a. (4)若以A为坐标原点, 分别以AB,AD,AA1所在的直线为x轴、y轴、z轴, 建立如图所示的空间直角坐标系, ∵正方体A1B1C1D1-ABCD的棱长为a, ∴C(a,a,0),C1(a,a,a).
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