已知:正四棱柱ABCD—A
1B
1C
1D
1中,底面邊長為2

,側(cè)棱長為4,E、F分別為棱AB、BC的中點.
(1)求證:平面B
1EF⊥平面BDD
1B
1;
(2)求點D
1到平面B
1EF的距離.
(1)證明略 (2)

(1) 建立如圖所示的空間直角坐標(biāo)系,則D(0,0,0),

B(2

,2

,0),E(2

,

,0),
F(

,2

,0),D
1(0,0,4),
B
1(2

,2

,4).

=(-

,

,0),

=(2

,2

,0),

=(0,0,4),
∴

·

=0,

·

=0.
∴EF⊥DB,EF⊥DD
1,DD
1∩BD=D,
∴EF⊥平面BDD
1B
1.
又EF

平面B
1EF,∴平面B
1EF⊥平面BDD
1B
1.
(2) 由(1)知

=(2

,2

,0),

=(-

,

,0),

=(0,-

,-4).
設(shè)平面B
1EF的法向量為n,且n=(x,y,z)
則n⊥

,n⊥

即n·

=(x,y,z)·(-

,

,0)=-

x+

y=0,
n·

=(x,y,z)·(0,-

,-4)=-

y-4z=0,
令x=1,則y=1,z=-

,∴n="(1,1,-"

)
∴D
1到平面B
1EF的距離
d=

=

=

.
練習(xí)冊系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖1,直角梯形

中,

,

,

,點

為線段

上異于

的點,且

,沿

將面

折起,使平面

平面

,如圖2.
(1)求證:

平面

;
(2)當(dāng)三棱錐

體積最大時,求平面

與平面

所成的銳二面角的余弦值.

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,在棱長為1正方體ABCD-A
1B
1C
1D
1中,M和N分別為A
1B
1和BB
1的中點
(1)求直線AM和CN所成角的余弦值;
(2)若P為B
1C
1的中點,求直線CN與平面MNP所成角的余弦值;
(3)P為B
1C
1上一點,且

,當(dāng) B
1D⊥面PMN時,求

的值.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖直角梯形OABC中,

,SO=1,以O(shè)C、OA、OS分別為
x軸、
y軸、
z軸建立直角坐標(biāo)系O-
xyz.
(Ⅰ)求

的大�。ㄓ梅慈呛瘮�(shù)表示);
(Ⅱ)設(shè)

①

②OA與平面SBC的夾角

(用反三角函數(shù)表示);
③O到平面SBC的距離.
(Ⅲ)設(shè)

①
.
②異面直線SC、OB的距離為
.
(注:(Ⅲ)只要求寫出答案).

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖3,直三棱柱

中,底面是等腰直角三角形,

,側(cè)棱

分別是

與

的中點,點

在平面

上的射影是

的重心

,求點

到平面

的距離.

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖所示,平行六面體ABCD—A
1B
1C
1D
1中,以頂點A為端點的三條棱長度都為1,且兩
兩夾角為60°.
(1)求AC
1的長;
(2)求BD
1與AC夾角的余弦值.

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
兩不重合直線l
1和l
2的方向向量分別為
=(1,0,-1),
=(-2,0,2),則l
1與l
2的位置關(guān)系是______.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
在

類比此性質(zhì),如下圖,在四面體P-ABC中,若PA、PB、PC兩兩垂直,底面ABC上的高為h,則得到的正確結(jié)論為__________________________.

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
若

,

,

是平面

內(nèi)的三點,設(shè)平面

的法向量

,則

________________。
查看答案和解析>>