则A1(a,0,a),D1(0,0,a),A(a,0,0),B(a,a,0),B1(a,a,a),E(a"," a"," a/2),F(0"," a/2 "," 0).

设平面A1D1E的一个法向量为n=(x,y,z),

则n·(A_1 D_1 ) ⃗=0,n·(A_1 E) ⃗=0,

即{■("(" x"," y"," z")·(-" a"," 0"," 0")" =0"," @"(" x"," y"," z")·" (0"," a",-" a/2)=0"," )┤

∴-ax=0,ay-a/2 z=0.

∴{■(x=0"," @y=z/2 "," )┤

令z=2,得n=(0,1,2).

直线GF到平面A1D1E的距离即为点F到平面A1D1E的距离,

又(FD_1 ) ⃗=(0",-" a/2 "," a),

∴所求距离d=("|" (FD_1 ) ⃗"·" n"|" )/("|" n"|" )=(3/2 a)/√5=(3√5)/10 a.

6.如图所示,在直三棱柱ABC-A1B1C1中,∠ABC=90°,BC=2,CC1=4,EB1=1,D,F,G分别为CC1,B1C1,A1C1的中点,EF与B1D相交于点H.

(1)求证:B1D⊥平面ABD;

(2)求证:平面EGF∥平面ABD;

(3)求平面EGF与平面ABD的距离.

(1)证明如图所示,建立空间直角坐标系,设A1(a,0,0),则C1(0,2,0),F(0,1,0),E(0,0,1),A(a,0,4),B(0,0,4),D(0,2,2),G(a/2 "," 1"," 0).