长方体ABCD-A
1B
1C
1D
1,其左视图沿AB方向投影,左视图如图.
(1)证明:AC
1⊥B
1C;
(2)当AC
1长为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898016_ST/0.png)
时,求多面体B
1-ABC
1D
1的体积.
考点分析:
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关于图中的正方体ABCD-A
1B
1C
1D
1,下列说法正确的有:
.
①P点在线段BD上运动,棱锥P-AB
1D
1体积不变;
②一个平面α截此正方体,如果截面是三角形,则必为锐角三角形;
③一个平面α截此正方体,如果截面是四边形,则此四边形必有一边平行;
④平面α截正方体得到一个六边形(如图),则截面α在平面AB
1D
1与平面BDC
1间平行移动时此六边形周长先增大,后减小.
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球放在墙角(两墙面,地面分别两两垂直),紧靠墙面和底面,墙角顶点到球面上的点的最远距离是
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898014_ST/0.png)
,则球的体积是
.(半径为R的球体积公式:
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898014_ST/1.png)
)
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如图,平面直角坐标系中,A(
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898013_ST/0.png)
,2),B(-
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898013_ST/1.png)
,-
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898013_ST/2.png)
),将其所在纸面沿x轴折成直二面角,则折起后的A,B两点的距离是
.
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由空间向量基本定理可知,空间任意向量
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898012_ST/0.png)
可由三个不共面的向量
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898012_ST/1.png)
唯一确定地表示为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898012_ST/2.png)
,则称(x,y,z)为基底
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898012_ST/3.png)
下的广义坐标.特别地,当
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898012_ST/4.png)
为单位正交基底时,(x,y,z)为直角坐标.设
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898012_ST/5.png)
分别为直角坐标中x,y,z正方向上的单位向量,则空间直角坐标(1,2,3)在基底
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898012_ST/6.png)
下的广义坐标为
.
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我们知道,在平面直角坐标系中,方程
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898011_ST/0.png)
表示的图形是一条直线,具有特定性质:“在x轴,y轴上的截距分别为a,b”;类比到空间直角坐标系中,方程
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898011_ST/1.png)
表示的点集对应的图形也具有某特定性质,设此图形为m,若m与zoy平面所成角正弦值为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898011_ST/2.png)
,则正数λ的值是( )
A.
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898011_ST/3.png)
B.
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898011_ST/4.png)
C.
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131103101936722589882/SYS201311031019367225898011_ST/5.png)
D.
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