(1) 以D为原点,DA、DC、AA1所在直线为X、Y、Z轴建立空间直角坐标系. D(0,0,0),B(1,1,0) D1(0,0,2),E(0,1,1),F( , ,1) ∴ =(1,1,0), =(0,0,2), =( ,- ,0) 由 · =0, · =0, 得,EF⊥DB,EF⊥DD1 ∴EF⊥面D1DB1---------------------------------------------------- (2) 设 =(x,y,z)是平面BDE的法向量, =(1,1,0), =(0,1,1) 由 ⊥ , ⊥ 得 即![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021003113-97420.png) ∴取y=1, =(-1,1,-1)
,由(2)知点 到平面BDE的距离为 = ---- (3) =(-1,-1,2) 由(2)知![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021003115-39063.png) 设直线BD1与平面BDE所成的角的正弦值为 ,则sin = ,cos =![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021003115-29585.png) ∴直线BD1与平面BDE所成的角的余弦值为 -------------------- |