1 . 已知E,F,G,H分别是空间四边形ABCD的边AB,BC,CD,DA的中点.
![](https://img.xkw.com/dksih/QBM/2021/12/30/2883785260703744/2885873556455424/STEM/1c8f5b2d-aaa7-494e-86b5-cbcfa9ab717e.png?resizew=234)
(1)用向量法证明E,F,G,H四点共面;
(2)设M是EG和FH的交点,求证:对空间任一点O,有
.
![](https://img.xkw.com/dksih/QBM/2021/12/30/2883785260703744/2885873556455424/STEM/1c8f5b2d-aaa7-494e-86b5-cbcfa9ab717e.png?resizew=234)
(1)用向量法证明E,F,G,H四点共面;
(2)设M是EG和FH的交点,求证:对空间任一点O,有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec1696ba7e9fc3f3b9837032c87f7fc8.png)
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2 . 在正四面体OABC中,E,F,G,H分别是OA,AB,BC,OC的中点.设
,
,
.
(1)用
,
,
表示
,
;
(2)用向量方法证明:E,F,G,H四点共面.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14390e9b6b44472bdc7a131133ab39b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87cd14dfc0024459f9d8e594c95c5106.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e07dcf0b16163e0e0e0c0f248466ee7e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/8/3d7b3045-39ae-4d68-a19b-b247708dab16.png?resizew=189)
(1)用
![](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/73a0b19e69be46452425916a0fcb49c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae54940f33b8714da5fe3b7546f8b3dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e74c0f207612e015857b78b99db483e4.png)
(2)用向量方法证明:E,F,G,H四点共面.
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2023-10-20更新
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2卷引用:陕西省西安市周至县第六中学2023-2024学年高二上学期10月月考数学试题
名校
解题方法
3 . 如图,在斜四棱柱
中,底面
是边长为1的正方形,
,记
.
![](https://img.xkw.com/dksih/QBM/2023/10/10/3342973715734528/3344512309739520/STEM/07d74803f6384d7c97e7de6c3528746d.png?resizew=179)
(1)证明:
;
(2)求侧棱
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e42d2046a0e800a6bf082172027612e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dfb9769a14ebf5cbc5fa0c06ce96435.png)
![](https://img.xkw.com/dksih/QBM/2023/10/10/3342973715734528/3344512309739520/STEM/07d74803f6384d7c97e7de6c3528746d.png?resizew=179)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aa142bb96af98b846997e681609739f.png)
(2)求侧棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
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2023-10-12更新
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5卷引用:湖南省部分学校(桃江县第一中学等校)2023-2024学年高二上学期10月联考数学试题
名校
4 . 在四棱柱
中,
,
.
(1)当
时,试用
表示
;
(2)证明:
四点共面;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/315fa3b12ca399d33a5033664671c494.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6471daa1b959fad099c6dfe471a1c6de.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/1/31639753-bf88-4555-be2d-46f9fbdddee8.png?resizew=154)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f2f91aa5dea19712561c7905535d15b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d68873c59a21b0cd408cdf2b47d51096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5c3accb1b8a5479439beff4259660e3.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c2d86d8daea5e652d99fe1c6bc3f9a.png)
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2023-09-01更新
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3卷引用:四川省成都市第七中学2023-2024学年高三上学期入学考试理科数学试题
5 . 如图,在四棱锥
中,四边形
为矩形,且
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/2/38ae0c01-7683-49ad-9e13-5a8b91d2cf89.png?resizew=175)
(1)求线段
的长度;
(2)求异面直线
与
所成角的余弦值;
(3)若
为
的中点,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80f51c31583fea58fde645474d60b8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b80ee363635d73f601654339028daec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e598f65e3c3b3c1a047a575788baee94.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/2/38ae0c01-7683-49ad-9e13-5a8b91d2cf89.png?resizew=175)
(1)求线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
(2)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e253f0fbfa13dca5b4c7dce45fc47fe7.png)
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2023-01-01更新
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4卷引用:安徽省滁州市定远县民族中学2022-2023学年高三上学期12月月考数学试题
安徽省滁州市定远县民族中学2022-2023学年高三上学期12月月考数学试题(已下线)模块一 专题11 空间向量与立体几何(已下线)第07讲 空间向量的数量积运算9种常见考法归类(1)河南省郑州市第一〇二高级中学2023-2024学年高二上学期10月月考数学试题
6 . 如图,在平行六面体
中,
,
,
,M,N分别为
,
中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/9/1e045548-b9c3-4a6d-b891-b810b9e2c4f6.png?resizew=185)
(1)求
的长;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5441d73845911db1993bf903c4d8700f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f3e58edd1f900ca82bb2a3058293f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6e7ac80553cc0af403a61741f3e351b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fbafedc202bd0d86c4dfdece9f8f4fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab35850dbc661ded6456b70767cc6cd0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/9/1e045548-b9c3-4a6d-b891-b810b9e2c4f6.png?resizew=185)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72f88e1c3e5bb643bafcda8f20d6a9a7.png)
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解题方法
7 . 在正四面体
中,
分别是
的中点.设
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc531bab94eb4396309eaa2065e8fffb.png)
![](https://img.xkw.com/dksih/QBM/2022/12/5/3124330800799744/3124500472496128/STEM/ddef02310f654657886823b3aba24da3.png?resizew=163)
(1)用
表示
;
(2)用向量方法证明;
①
;
②
四点共面.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c2d86d8daea5e652d99fe1c6bc3f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae00ee1e21e0931ce9a73afafa4d832f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/420ad6159fc091d6a5ffddf0676d2662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc531bab94eb4396309eaa2065e8fffb.png)
![](https://img.xkw.com/dksih/QBM/2022/12/5/3124330800799744/3124500472496128/STEM/ddef02310f654657886823b3aba24da3.png?resizew=163)
(1)用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae951e0bb5a2a406f1572fc1e4964265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b217b80be94d29bb07778b7eac5344a6.png)
(2)用向量方法证明;
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c31c9f6ee41257798d7740a2043e108.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c2d86d8daea5e652d99fe1c6bc3f9a.png)
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解题方法
8 . 如图,在平行六面体
中,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/15/86de3ee0-071b-4c8e-b591-625fa67ee6b3.png?resizew=167)
(1)求
的长;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6539b62084f39afabcd9b93c2c562b7e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/15/86de3ee0-071b-4c8e-b591-625fa67ee6b3.png?resizew=167)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aa142bb96af98b846997e681609739f.png)
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9 . 如图所示,四面体
中,G,H分别是
的重心,设
,点D,M,N分别为BC,AB,OB的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/23/4d160c3a-249c-4cb4-a5de-8cdf8321b8c7.png?resizew=243)
(1)试用向量
表示向量
;
(2)试用空间向量的方法证明MNGH四点共面.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1262ec403745d82befa99d4c6c2ae35b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d8ab14cb4f1a62f730d56f702f6e99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87f570823f32dce24caed626e00a0857.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/23/4d160c3a-249c-4cb4-a5de-8cdf8321b8c7.png?resizew=243)
(1)试用向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3865341eda32747025e067ad4cc17ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f54bf88a2dfb8265280b9d07e7ee528.png)
(2)试用空间向量的方法证明MNGH四点共面.
您最近一年使用:0次
2022-10-20更新
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761次组卷
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7卷引用:河北省沧州市部分学校2022-2023学年高二上学期第一次阶段测试数学试题
河北省沧州市部分学校2022-2023学年高二上学期第一次阶段测试数学试题(已下线)6.1.3 共面向量定理(练习)-2022-2023学年高二数学同步精品课堂(苏教版2019选择性必修第二册)(已下线)6.1.3共面向量定理(1)1.1.1 空间向量及其线性运算练习福建省福州市山海联盟教学协作校2023-2024学年高二上学期期中联考数学试题(已下线)专题01空间向量及其运算(4个知识点8种题型3个易错点)(3)(已下线)6.1 空间向量及其运算(4)
10 . 正多面体也称柏拉图立体,被誉为最有规律的立体结构,是所有面都只由一种正多边形构成的多面体(各面都是全等的正多边形).数学家已经证明世界上只存在五种柏拉图立体,即正四面体、正六面体、正八面体、正十二面体、正二十面体.已知一个正八面体ABCDEF的棱长都是2(如图),P,Q分别为棱AB,AD的中点,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3350049b484df2df02602524fa047c6.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3350049b484df2df02602524fa047c6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/21/b5ade79c-80e9-4575-aa6a-f05148f91559.png?resizew=145)
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2022-09-19更新
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10卷引用:河南省创新联盟2022-2023学年高二上学期第一次联考(B卷)数学试题
河南省创新联盟2022-2023学年高二上学期第一次联考(B卷)数学试题湖北省襄阳市部分学校2022-2023学年高二上学期9月联考数学试题湖北省孝感市部分学校2022-2023学年高二上学期9月联考数学试题(已下线)第04讲 空间向量在立体几何中的应用(练,理科专用)湖北省襄阳市第二中学2022-2023学年高二上学期9月月考数学试题辽宁省沈阳市第十中学2022-2023学年高二上学期10月月考数学试题(已下线)专题16 空间向量及其应用(练习)-1(已下线)模块二 专题1 《空间向量与立体几何》单元检测篇 B提升卷(苏教 )辽宁省沈阳市第十中学2022-2023学年高二上学期第一阶段考试数学试题(已下线)1.2 空间向量基本定理【第三练】