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1 . 下列选项中,不正确的命题是( )
A.若两条不同直线![]() ![]() ![]() ![]() ![]() |
B.若![]() ![]() ![]() ![]() ![]() ![]() |
C.若![]() ![]() |
D.若空间向量![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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2024-03-19更新
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599次组卷
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3卷引用:辽宁新高考联盟(点石联考)2023-2024学年高二下学期3月联合考试数学试题
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2 . 如图,在空间直角坐标系
中,正四棱柱
的底面边长为4,高为2,O为上底面中心,E,F,G分别为棱
、
、
的中点.若平面
与平面
的交线为l,则l的方向向量可以是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76ee17afa983fa795545b5568b80089.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d42170c7d4249f6b390823606c18c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a0f9bc9123d19a09babe8609cf12327.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80cc8e81f6bd1e28869e0a46e1732320.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/2/d70f2417-2189-44bb-80e6-0a98b9c03880.png?resizew=197)
A.![]() | B.![]() | C.![]() | D.![]() |
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3 . 如图,在四面体
中,
分别是
的中点,
是
和
的交点,
为空间中任意一点,则( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/15/128f243b-03d0-451d-867e-824ddbd8d002.png?resizew=162)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f68f2c6d854d7ca94f77c0c9ba969dd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f758a811b8a2c260431aa7e6a4060987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83f1f880e5ffbff036953acaca90c41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/15/128f243b-03d0-451d-867e-824ddbd8d002.png?resizew=162)
A.![]() |
B.![]() |
C.![]() ![]() |
D.![]() |
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4 . 关于空间向量,以下说法正确的是( )
A.若非零向量![]() ![]() ![]() ![]() ![]() ![]() |
B.若对空间中任意一点O,有![]() |
C.若空间向量![]() ![]() ![]() ![]() ![]() |
D.已知直线l的方向向量为![]() ![]() ![]() ![]() ![]() |
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2024-02-04更新
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263次组卷
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2卷引用:河北省保定市定州市2023-2024学年高二上学期1月期末考试数学试题
5 . 如图,四棱柱ABCD-A1B1C1D1为正方体,
①直线DD1的一个方向向量为
;
②直线BC1的一个方向向量为
;
③平面ABB1A1的一个法向量为
;
④平面B1CD的一个法向量为
;
则上述结论正确的是___________ (填序号)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/17/19929bb0-66c9-4dd8-9006-8f10acb6d4be.png?resizew=167)
①直线DD1的一个方向向量为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe23a9eeae59b3fdf1302b1d22301ad5.png)
②直线BC1的一个方向向量为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/068b7ce272ebb01cb679bcdbabf64aa1.png)
③平面ABB1A1的一个法向量为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52de1861d246ea4a2e8bcf36315de77b.png)
④平面B1CD的一个法向量为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c37c96c0e0b232eed99ae5e300ea1e6c.png)
则上述结论正确的是
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解题方法
6 . 类似平面解析几何中的曲线与方程,在空间直角坐标系中,可以定义曲面(含平面)
的方程,若曲面
和三元方程
之间满足:①曲面
上任意一点的坐标均为三元方程
的解;②以三元方程
的任意解
为坐标的点均在曲面
上,则称曲面
的方程为
,方程
的曲面为
.已知曲面
的方程为
.
过曲面
上一点
,以
为方向向量,求证:直线
在曲面
上(即
上任意一点均在曲面
上);
(2)已知曲面
可视为平面
中某双曲线的一支绕
轴旋转一周所得的旋转面;同时,过曲面
上任意一点,有且仅有两条直线,使得它们均在曲面
上.设直线
在曲面
上,且过点
,求异面直线
与
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fa13f27f92b55bd7ecd9d750f98ae99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eedd047376c4cf1b9992cd8e4fe20df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d7a5b312e3d789a1070000315d63b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0b4e60b50889c152c56fcf2d6ea35bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知曲面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7d96461d2b3421aed548b754637ca8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44ddad9072eb1393ce15ca0627c941b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
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7 . 在空间直角坐标系中,已知向量
,点
,点
.若直线l经过点
,且以
为方向方量,P是直线l上的任意一点,O为坐标原点.
(1)求证:
;
(2)当
,且
时,求点P的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba436b158708e7325885fea7106e92b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c832b5312310a88bef6596496df8daa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bfcaf2a345411411cf94422703e9269.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9f50605db5d5f8f3a01ee8e474a112.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d71d1e5f816103a951d6ebf10af047b.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1f0582d9908f92f14cb02a6ccaf0eae.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1965236b071589fcdd33364bb0a92d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db835cce1e68a580450593b3905e5d8.png)
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解题方法
8 . 空间直角坐标系
中,经过点
且法向量为
的平面方程为
,经过点
且一个方向向量为
的直线l的方程为
,阅读上面的材料并解决下列问题:现给出平面α的方程为
,经过点
的直线l的方程为
,则直线l与平面α所成角的正弦值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e336d6ca2cae3d6e6c3810d7e521a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baf95be25d34a7366bf4060d081329c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d4ac7c3e4d50ea4683f54f80f23ffff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff4c767999174c06cee1c45b8fa908d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab9f353152c7f589c0caf5f964f803ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/181d60545ee1d7000e1138709be1b4bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b335c281e58bcfc69ed9e9feea4420a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abbbfd8b995cf3ad0cb7667f4419ddc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b760ad4c3ac6422cc6f771093ec8ffbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0b1e0c3fcb578008f60e4892db10260.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2023-12-25更新
|
423次组卷
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2卷引用:四川省成都市西北中学2023-2024学年高二上学期12月阶段性检测数学试题
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9 . 在长方体
中,
为长方体表面上的动点,且
,则点
的轨迹的长度为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9d4028faa379bbaa5d1e3be29177a46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d49e349a2d25b22bdb5384d0b733bb2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
A.2 | B.![]() | C.![]() | D.![]() |
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10 . 已知点
,
,
,
,过点P作
平面OAB,H为垂足,则点H的坐标是_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d560631b4b30e9b4cf3277c7f7cce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25261fd910651e83a70d5b91f0c5d45b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dcf4cc47c33024eb16e7acdd6ff92f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a70c639329bbf35173827da8d8c20e85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b17622ea6f6f5afd1ad817a557e5889d.png)
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