2024高三·全国·专题练习
1 . 如图,在平行六面体ABCD-A1B1C1D1中,设AA1=a,=b,
=c,M,N,P分别是AA1,BC,C1D1的中点.试用a,b,c表示以下各向量:
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7239b3f2d88c2e45e17e5de9ae1a332.png)
(2)A1N;
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76782c6299c144c919de3213c60a7e5f.png)
您最近一年使用:0次
2024高三·全国·专题练习
2 . 如图平行六面体
中,三棱
间夹角均为
.求:
(1)对角线
的长度;
(2)平行六面体全面积
;
(3)平行六面体体积
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d92178a5c6a76586292f622da298a8db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a12ffb2088a13eda785cb147122375e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/21/5a0bdf7f-a881-4e5c-8746-4fecfe952220.png?resizew=175)
(1)对角线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
(2)平行六面体全面积
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69e5c11ced8d1483c518ccd650924e47.png)
(3)平行六面体体积
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
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名校
3 . 三阶行列式是解决复杂代数运算的算法,其运算法则如下:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c971b2ecdfce17d75d0290dd194baa3b.png)
.若
,则称
为空间向量
与
的叉乘,其中
,
,
为单位正交基底.以
为坐标原点,分别以
的方向为
轴、
轴、
轴的正方向建立空间直角坐标系,已知
是空间直角坐标系中异于
的不同两点.
(1)①若
,求
;
②证明:
.
(2)记
的面积为
,证明:
;
(3)问:
的几何意义表示以
为底面、
为高的三棱锥体积的多少倍?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c971b2ecdfce17d75d0290dd194baa3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b759f4d0af0d28b35bdd5648db70968.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b44b6a86302386ebf96b784d02b039c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b4e6bae1b67a0a1eeafdd1114a792df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0f4837cd4b882c0380201dd437e7ae1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2772831f709c3c7c9a334b9444e0504.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/808173f5aafa97a38056d68247d68314.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4664eed9e1abab0ed6397c58d70e731.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(1)①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faea453a5148e6b281c75a0caa793452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00db2bada2cfc90c5213aca8af17df4c.png)
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f36900d061dee46d3f76344ac576ba1.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29aa828f2bd9a5e63ee58dcaa9d0d336.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd55f2f03192e5f0d76bf1cdb51872f2.png)
(3)问:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db0cfd110195cf5e453947d1648ef605.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4124e7ab7a93ee45858b3a4d4ab3508b.png)
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2024-03-26更新
|
636次组卷
|
3卷引用:压轴题04立体几何压轴题10题型汇总-2
2024高三·全国·专题练习
解题方法
4 . 如图,四棱锥
中,底面
为直角梯形,
,
,设
的二面角为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://img.xkw.com/dksih/QBM/2024/3/19/3456860575956992/3456980262281216/STEM/5afb8017eae245c6bd74fd0e06528f53.png?resizew=183)
(1)当
时,求
的体积;
(2)设N为
的中点,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb3f9a5da641be35117fd35ba07a6aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f571396be1aa4a8914a66f7d7abd6381.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d52bebecac3bc1e5901a99a5ab23fe97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d294d69caac577339f11f477b2047e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://img.xkw.com/dksih/QBM/2024/3/19/3456860575956992/3456980262281216/STEM/5afb8017eae245c6bd74fd0e06528f53.png?resizew=183)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a10f3e254ade296ead255d9f463baee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
(2)设N为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ffb98f1e3c1317c0db403d3af04bdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38aae33d8282c6fbb86fc9bae315cbee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa87e8c74812374d20ed902d25d3d918.png)
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2024高二上·江苏·专题练习
5 . 已知
,
,
是三个不共面的向量,
,
,
,且
,
,
,
四点共面,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e73db31aecdde14e0002f082d9091df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2980a18e4d0a2a795b7983a1a1866db7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1821c677712026f8de34fe924b1f52a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b30ac02890469b4716c89e9005e9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a957b6ad6202a90de7da0d61d39f6884.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d79639c1de0a711e8747f2535f1395e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
名校
解题方法
6 . 三阶行列式是解决复杂代数运算的算法,其运算法则如下:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/281c685c6a79c8293c8b5083c3a8dc4c.png)
若
,则称
为空间向量
与
的叉乘,其中
,
,
为单位正交基底. 以
为坐标原点、分别以
,
,
的方向为
轴、
轴、
轴的正方向建立空间直角坐标系,已知
,
是空间直角坐标系中异于
的不同两点
(1)①若
,
,求
;
②证明
.
(2)记
的面积为
,证明:
.
(3)证明:
的几何意义表示以
为底面、
为高的三棱锥体积的
倍.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/281c685c6a79c8293c8b5083c3a8dc4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e91aaddb8691f8afa477a96bf630631.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aba64ae92194bc4f0f6e49725471542.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8643f24c3af715421ec0ccd3224ed453.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d541143135cb9b8166bc631a85ac6a41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a471332d4f3731d90f62fdf819f39824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e73db31aecdde14e0002f082d9091df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2980a18e4d0a2a795b7983a1a1866db7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1821c677712026f8de34fe924b1f52a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(1)①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41ef077626c88964805a45849471a1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bb22d1c614d99e2639864e43f4b6277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00db2bada2cfc90c5213aca8af17df4c.png)
②证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb8623a42db5ceb745a16d72739f513.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29aa828f2bd9a5e63ee58dcaa9d0d336.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0505ce82dd5726c22fcaac54d01d630b.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8191a760981f2d67648905665c8b167a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad58b362528b814739ceb7fe5febfc28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
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2024-03-07更新
|
902次组卷
|
8卷引用:专题22 新高考新题型第19题新定义压轴解答题归纳(9大核心考点)(讲义)
(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大核心考点)(讲义)(已下线)第七章 应用空间向量解立体几何问题拓展 专题二 平面法向量求法及其应用 微点2 平面法向量求法及其应用(二)【培优版】河南省部分重点高中2024届高三普通高等学校招生全国统一考试(期末联考)数学试卷江苏省扬州市仪征中学2024届高三下学期期初调研测试数学试题 河南省部分重点高中(青桐鸣)2023-2024学年高三上学期期末大联考数学试题江苏省江都中学2023-2024学年高二下学期3月联考数学试卷江苏省盱眙中学2023-2024学年高二下学期第一次学情调研数学试题(已下线)专题08 期末必刷解答题专题训练的7种常考题型归类-期末真题分类汇编(北师大版2019必修第二册)
2024高二上·全国·专题练习
7 . 如图所示,已知点P为正方形ABCD所在平面外一点,且
平面ABCD,M,N分别是AB,PC的中点,且
,求向量
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f6967901d6c855864df01e7bf7a15c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a014dff8997c661055229de29c61cfc.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/18/f9e6291a-7221-4928-8be7-26e0c05682ae.png?resizew=144)
您最近一年使用:0次
2024高二·全国·专题练习
解题方法
8 . 如图,
⊥
,
⊥
,
⊥
,
,
分别是
的中点,
分别是
的中点,证明:
⊥
.
![](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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5969d2c05955f48cd5b3b65629ad5151.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2013303974d530fce9a9930938f4c2b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/16/0cb92a8e-69a6-494a-9fab-fc5108a2b1fd.png?resizew=171)
您最近一年使用:0次
9 . 如图,四面体
的所有棱长均为2,D,F分别为
,
的中点,且点E为
的三等分点(靠近点B).
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/11/1e969929-e6e1-422d-8245-f4856eea71f3.png?resizew=160)
(1)设向量
,
,
,用
,
,
表示向量
;
(2)求点D到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/334a5773c8d24f29ec3231075170e4f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c77e9c89b7275b0c1a9af5c9a72e5968.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/11/1e969929-e6e1-422d-8245-f4856eea71f3.png?resizew=160)
(1)设向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fbd0a772a009eec08ec6d50a59eae67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7824b7ecd7d62f8a76a05c88bfd6693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95e02cdf7ad2557cd1e92a8b438a9f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d366d8fbb7258ee051f49977441e14a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa8c53645db602c72b00b599c2c0ff97.png)
(2)求点D到平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3747e528a1e8d45668ccf835c0175a73.png)
您最近一年使用:0次
2024-01-22更新
|
138次组卷
|
4卷引用:第七章 应用空间向量解立体几何问题拓展 专题一 空间向量基底法 微点3 空间向量基底法(三)【基础版】
(已下线)第七章 应用空间向量解立体几何问题拓展 专题一 空间向量基底法 微点3 空间向量基底法(三)【基础版】四川省巴中市2023-2024学年高二上学期期末考试数学试卷四川省遂宁市2023-2024学年高二上学期期末质量监测数学试题四川省雅安市2023-2024学年高二上学期期末教学质量检测数学试题
2024高三·全国·专题练习
解题方法
10 . 如图,在正方体
中,
,
,
,点M,N分别是
,
的中点.
(1)试用
,
,
表示
.
(2)求证:
平面
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a08f3a88dffed011df93d1d606a08ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f8173bb7787b6b107acfe767dd1d91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/232311b4261c36b659555a07bfa00f7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cfbc0b5a8fbde804bd8425a4b76d207.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/3/d1ed7a47-813c-483a-a098-5d06cabb43a5.png?resizew=171)
(1)试用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d366d8fbb7258ee051f49977441e14a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a014dff8997c661055229de29c61cfc.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7592c4f01c8e06c7ee90df5b9413a9f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
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