在正方体ABCD-A1B1C1D1中,E,F分别是DD1,BD的中点,G在棱CD上,且CG=1⼀4

CD,H为C1D的中点 应用空间向量法求证 EF垂直B1C
2025-05-19 14:25:46
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回答1:

建立坐标系D(0,0,0),A(1,0,0),C(0,1,0),D1(0,0,1)

E(0,0,1/2),F(1/2,1/2,0),G(0,3/4,0), H(0,7/8,1/2), C1(0,1,1)

EF = (1/2,1/2,-1/2), C1G = (0,-1/4, -1), cos(EF-C1G) = (-1/8-1/2)/根号(3/4 * 17/16)=5/根号(51)

FH = 根号(1/4 + 9/64 + 1/4) = 根号(41)/8