1 . 在三棱锥中,
和
都是等边三角形,
,
,
为棱
上一点,则
的最小值是
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2 . 如图,设
为
所在平面外任意一点,
为
的中点,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edff1881635893293dd411ead8194aca.png)
_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc840e197de2dabff676ee1b8968fada.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be8c9a4f4f4a64ba852a1346c10113ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edff1881635893293dd411ead8194aca.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/27/5c8fb242-2ca6-45f2-af26-ac1937778622.png?resizew=142)
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3 . 如图,三棱锥O-ABC中,M是BC的中点,
,设
用
表示向量
则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bf61a6de70565fbf32abe3c53c79312.png)
_____
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6e979e750bebbabd2a5fc706de928f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c515266d44d063e3d18da3bc556f293.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e12e95f703ad30ab9a3d38376830989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2490b8c788fcdec42cb81fae32f9d0e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bf61a6de70565fbf32abe3c53c79312.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/20/cd013d69-34b6-4b0f-8bc3-6f0b486d43ce.png?resizew=146)
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4 . 如图,在三棱柱
中,D,E分别是线段
,
的中点,设
,
,
.用
,
,
表示![](https://staticzujuan.xkw.com/quesimg/Upload/formula/658ff9921ff96972b0fc95360bb0c65b.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0879a90fdf1c8be67eccfff8fe2c4cdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9206f052cc7298fdf9049f9386f53928.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df107a0db8e7084e46cbf0edc9dd4878.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/73a0b19e69be46452425916a0fcb49c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/658ff9921ff96972b0fc95360bb0c65b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/7/6d5f40ae-63d4-44bf-8303-0e9a5045899f.png?resizew=151)
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名校
5 . 已知三棱锥
,空间内一点
满足
,则三棱锥
与
的体积之比为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9de695c560aeff95fdde7d0b4476ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5d90f940f5693b22ddf2e7c761887d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
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2024-04-07更新
|
759次组卷
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4卷引用:山东省齐鲁名校联盟2023-2024学年高三第七次联考数学试题
6 . (6)已知空间中
两点,则
两点之间的距离公式为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c84a09595eabd6ae21a2e8faa7465c.png)
_______________
(7)在空间直角坐标系中,y轴上的点的坐标形式为___________
(8)向量加减法运算法则:加法三角形法则:首尾相连,首指向尾为和.
加法平行四边形法则:共起点的两边为邻边作平行四边形,共起点_________ 为和.
减法三角形:同起点,连终点,方向________ .
(9)共线向量基本定理:空间两个向量![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b172cf8d898883d82e973f28c3c3a3e.png)
共线的充要条件是存在唯一的实数
,使得__________ .通常把这个定理称为共线向量基本定理.
(10)数乘运算律:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294c0f85a47e8f5e004611af0bdd75a5.png)
_______________ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93a4ee3c424455ab04b1cf0acfb87d62.png)
____________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3557f7de9f773172e5cd05574e958cbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c84a09595eabd6ae21a2e8faa7465c.png)
(7)在空间直角坐标系中,y轴上的点的坐标形式为
(8)向量加减法运算法则:加法三角形法则:首尾相连,首指向尾为和.
加法平行四边形法则:共起点的两边为邻边作平行四边形,共起点
减法三角形:同起点,连终点,方向
(9)共线向量基本定理:空间两个向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b172cf8d898883d82e973f28c3c3a3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4220bc106912baca47334a7b767cb851.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(10)数乘运算律:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294c0f85a47e8f5e004611af0bdd75a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93a4ee3c424455ab04b1cf0acfb87d62.png)
您最近一年使用:0次
7 . 有下列命题:
①若
,则
四点共线;
②若
,则
三点共线;
③若
为不共线的非零向量,
,则
;
④若向量
是三个不共面的向量,且满足等式
,则
.
其中是真命题的序号是________ (把所有真命题的序号都填上).
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32458db21db3e131308b31f48353fbb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de2c38323135e90ae80394717f3de5cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
③若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fb4f795474089c4ca5183f0b8c8210d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38c918fe6a20db9937e1bc36dfaa0eae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91b958a367fa2f08b6202a5a6ebf5e9f.png)
④若向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fb4f795474089c4ca5183f0b8c8210d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0751e700350ac26971ec5e7df4ba6abd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ebddf666811739c8b11b35c5c339f19.png)
其中是真命题的序号是
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名校
解题方法
8 . 在正四棱台
中,
,
,
,
,
,若
平面
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5085e3cdef9ea6c564e079f745d6fdb.png)
_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6834ac70927ae08d7d36a1922403c9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa70bd6be0a8808b360996dbccbefdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92535536bd3c2761724fd058427f95a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a23f01af749100e1888bba06268843db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/213a62d7a1846c10fd66fc25e2afb0e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3a04ea8ebc597fd1f5d6bb8df181a2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fae7f4612c548b1f72a964ddb291cd2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5085e3cdef9ea6c564e079f745d6fdb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/26/302e09be-1d6c-43bd-bf9d-31ddc7bba802.png?resizew=177)
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2023-10-09更新
|
463次组卷
|
10卷引用:陕西省部分学校2023-2024学年高二上学期10月联考数学试题
陕西省部分学校2023-2024学年高二上学期10月联考数学试题山东省聊城第一中学2023-2024学年高二上学期10月月考数学试题山东省部分学校2023-2024学年高二上学期10月质量检测联合调考数学试题陕西省西安市灞桥区2023-2024学年高二上学期第一次联考数学试题吉林省部分名校2023-2024学年高二上学期10月联考数学试题山东省泰安市肥城市第一高级中学2023-2024学年高二上学期10月月考数学试题陕西省部分学校(西安市第八十六中学等)2023-2024学年高二上学期10月联考数学试题(已下线)高二数学上学期期中模拟卷01(原卷版)(已下线)3.1 空间向量及其运算(八大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)(已下线)第八章 立体几何初步(二)(知识归纳+题型突破)(1)-单元速记·巧练(人教A版2019必修第二册)
名校
9 . 如图,空间四边形
的各边及对角线长都为2,
是
的中点,
在
上,且
,则向量
与向量
所成角的余弦值为______________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b70f6944425737555d92b73fedeaceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21909dd065ccc349a2cbfd4c3cf4976b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba7d3a0680780aaf4549c447fe8dfe9f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/8/60069ad8-ced5-4ded-b4b1-48c425ab4d80.png?resizew=185)
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2023-09-07更新
|
1056次组卷
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4卷引用:福建省南安市华侨中学2023-2024学年高二上学期8月月考数学试题
福建省南安市华侨中学2023-2024学年高二上学期8月月考数学试题重庆市石柱回龙中学校2023-2024学年高二上学期9月质量检测数学试题四川省遂宁市射洪绿然学校2023-2024学年高二上学期第一学月考试数学试题(已下线)专题01 空间向量的线性运算(考点清单)-2023-2024学年高二数学上学期期中考点大串讲(人教A版2019选择性必修第一册)
10 . 在三棱柱
中,点
在线段
上,且
,若以
为基底表示
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98ce0da153621bbd05519f168b527e3e.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fd4c85bb98a2a0afddd7ed92578ad2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b6285803e186abb615e9add52034bfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4b104831ce8f0486de0faf40d5b24d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe6d728b430549f00bb9c0a7bf8bf7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98ce0da153621bbd05519f168b527e3e.png)
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2023-06-29更新
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455次组卷
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3卷引用:江苏省淮安市2022-2023学年高二下学期期末数学试题