1 . 如图,
分别是四面体
的棱
的中点,
是
的三等分点(点
靠近点
),若
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/29/db3f5fea-2c8f-4f36-ba32-c648a8b64df6.png?resizew=176)
(1)以
为基底表示
;
(2)若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/881ec4ef24907683dd46062e5ff149c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2237240a3a5108783d1b90c0d7572806.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/29/db3f5fea-2c8f-4f36-ba32-c648a8b64df6.png?resizew=176)
(1)以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ed18a92338c7578c18a5ba3a2ae1ed4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4731335d26e45bf7041b36c5f0a1121d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0d27a9e2e35a727df29ff1d87d73dc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/694cdf60ef30ed1cede03a92e866d70c.png)
您最近一年使用:0次
2023-10-11更新
|
336次组卷
|
3卷引用:山西省运城市教育发展联盟2023-2024学年高二上学期10月调研测试数学试题
山西省运城市教育发展联盟2023-2024学年高二上学期10月调研测试数学试题辽宁省沈阳市重点学校联合体2023-2024学年高二上学期期末检测数学试题(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】
名校
2 . 四棱柱
的六个面都是平行四边形,点
在对角线
上,且
,点
在对角线
上,且
.
(1)设向量
,
,
,用
、
、
表示向量
、
;
(2)若
、
、
三点共线,求实数
的取值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e663220a66eff19da6a71e46b397db2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724a298d2319f26dc085f09b5169125c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eb97aff0960e2640314888a38e7169c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2fd9a92fe86ce127b606f01333f8986.png)
(1)设向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7b9d8a08fc52c31cc1a7f527d18b55c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/184359fe3cadc363cf4ebe586c2b3db4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39b0ff98fe5e0a913ebecda552acc6f6.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/afdb48133ed00fa1327ac46ca7f344b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7ef300fccd1d15cfd5556f9d742e12f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
3 . 如图,在平行六面体
中,
,
,设
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2410e0e103cc41264a2d0f5c0e3f3ed4.png)
(1)用
,
,
表示出
,并求线段
的长度;
(2)求直线
与
夹角的余弦值;
(3)用向量法证明直线
平面
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cead0e8eadfdcefa334953e88864f424.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4d13ecb54b1006051d2561327aa4755.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7b9d8a08fc52c31cc1a7f527d18b55c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/184359fe3cadc363cf4ebe586c2b3db4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2410e0e103cc41264a2d0f5c0e3f3ed4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/12/94413b45-3a00-4a8f-8038-d85b7ced15b5.png?resizew=122)
(1)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/053c0f6846f2bf8671b351a4263a0270.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b100adef1832f7236e74d6150629ac98.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fe734023d4e70010a6b2cc3267cb86e.png)
(3)用向量法证明直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d1d2e0f281222a5f289ea4008370aed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7935fe3125f247b7bea4f065ce9ad985.png)
您最近一年使用:0次
名校
4 . 空间中,两两互相垂直且有公共原点的三条数轴构成直角坐标系.如果坐标系中有两条坐标轴不垂直,那么这样的坐标系称为“斜坐标系”.现有一种空间斜坐标系,它任意两条数轴的夹角均为
,我们将这种坐标系称为“斜
坐标系”.我们类比空间直角坐标系,定义“空间斜
坐标系”下向量的斜
坐标:
分别为“斜
坐标系”下三条数轴(
轴,
轴,
轴)正方向上的单位向量,若向量
,则
与有序实数组
一一对应,称向量
的斜
坐标为
,记作
.
(1)若
,求
的斜
坐标;
(2)在平行六面体
中,
,建立“空间斜
坐标系”如下图所示.
①若
,求向量
的斜
坐标;
②若
,且
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4664eed9e1abab0ed6397c58d70e731.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.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/138c39673b579f1346c38398811105a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b8a88a16125366536cb4ad658e0cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee4e3cf72016a2b908b9178b8317b84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b8a88a16125366536cb4ad658e0cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee4e3cf72016a2b908b9178b8317b84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/971975007772deb92f837127a7936389.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9fe31c74115f017c61dc7e6d78d5fac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9cd8bbf47b69bbd7a6263b041290d11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
(2)在平行六面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a96b1713aa3c420848e9865afefa3fb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/27/30f606bc-c728-4798-a6d6-a33bb22e85bf.png?resizew=181)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/099edd3520292558184521a9af4e9064.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/253b99b8c8a45ace50b590cdd89b238a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bcc897340d513ba62e60b02e5ede30a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59cbb23e8edee78010195fe66d3e55b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5813dd9f2bd01a38d749247eccca5449.png)
您最近一年使用:0次
2023-10-10更新
|
183次组卷
|
3卷引用:湖北省宜荆荆随2023-2024学年高二上学期10月联考数学试题
名校
5 . 空间中,两两互相垂直且有公共原点的三条数轴构成直角坐标系,如果坐标系中有两条坐标轴不垂直,那么这样的坐标系称为“斜坐标系”.现有一种空间斜坐标系,它任意两条数轴的夹角均为60°,我们将这种坐标系称为“斜60°坐标系”.我们类比空间直角坐标系,定义“空间斜60°坐标系”下向量的斜60°坐标:
分别为“斜60°坐标系”下三条数轴(
轴、
轴、
轴)正方向的单位向量,若向量
,则
与有序实数组
相对应,称向量
的斜60°坐标为
,记作
.
(1)若
,
,求
的斜60°坐标;
(2)在平行六面体
中,
,
,N为线段D1C1的中点.如图,以
为基底建立“空间斜60°坐标系”.
①求
的斜60°坐标;
②若
,求
与
夹角的余弦值.
![](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/138c39673b579f1346c38398811105a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b8a88a16125366536cb4ad658e0cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b525d8c768efd801ab58bc4c0da9221e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b8a88a16125366536cb4ad658e0cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f60fd9ea272088c32da829aea1de070b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ad77af674bcbc49460fb989fa973372.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/28/529200b0-ed2f-4650-8678-cb630e8d7d0f.png?resizew=193)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ade1012bfb509cb44ee60d6111e439.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e1f037129b07c0be3c9be28929655bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9cd8bbf47b69bbd7a6263b041290d11.png)
(2)在平行六面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a698be6c34b89c748764041281fd4da2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f0b24d3b14c326b2baa2d2c5e8db871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be560befd3ac8e670f8b6edd15edf31d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81b462f38860b00ac3b9bb1708ddd7bd.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/350f162ee9aa08f4c9779481a5ef1025.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59c8938a2b0b3c9971764f833bb37a15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe6d728b430549f00bb9c0a7bf8bf7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/350f162ee9aa08f4c9779481a5ef1025.png)
您最近一年使用:0次
2023-10-10更新
|
947次组卷
|
7卷引用:四川省绵阳南山中学2023-2024学年高二上学期10月月考数学试题
四川省绵阳南山中学2023-2024学年高二上学期10月月考数学试题河南省实验中学2023-2024学年高二上学期期中考试数学试题(已下线)模块一 专题1 空间向量的基本运算 B提升卷 期末终极研习室(2023-2024学年第一学期)高二人教A版安徽省卓越县中联盟2024届高三上学期第三次质量检测数学试题四川省成都市2023-2024学年高二上学期期末练习数学试题(3)(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)江苏省南京人民中学、海安实验中学与句容三中2023-2024学年高二下学期3月月考数学试题
解题方法
6 . 已知在空间四边形
中,
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62d52be7c6e607972b4cf8ccbf58436.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22bebf68c92f3fc0de750330b001df56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846bbe6e9646cae074a24c0dffc6d8f3.png)
您最近一年使用:0次
7 . 在长方体
中,已知
,
,
.
的坐标;
(2)求
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7479255f54d51b97e6314db1dc06eb22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7d52fd1bdfe4ed2f1dc275b84a599e1.png)
您最近一年使用:0次
8 . 如图,在平行六面体
中,G为
的重心.设
,
,
,以
,
,
为一组基,求
,
在这组基下的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3ea25ef38e4afa8f75ffd0842890289.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f3adc4ed291596abf3bb93ae7a075d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e984585ddf28c039219afcebf229de7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a5ad65006ab94c402084227f4675b57.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/333a232d5882b2f03f9e02846c442a95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/538c1a9e933bc1e29d785eeda9cd1abc.png)
您最近一年使用:0次
9 . 如图,平行六面体
中,点M在线段
上,且
,点N在线段
上,且
.求证:M,N,
三点在一条直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e663220a66eff19da6a71e46b397db2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9bfbf0f38638236ea1b1a96ed04dee3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eb97aff0960e2640314888a38e7169c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71594be1602bece8a76509363cdbdff9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/22/b978ae31-3c45-4c68-98ee-63b91ba2b7cc.png?resizew=154)
您最近一年使用:0次
名校
10 . 如图所示,在平行六面体
中,
,
,
,P是
的中点,M是
的中点,N是
的中点,用基底
表示以下向量:
(1)
;
(2)
;
(3)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7b9d8a08fc52c31cc1a7f527d18b55c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/184359fe3cadc363cf4ebe586c2b3db4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39b0ff98fe5e0a913ebecda552acc6f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a037b1a3ec2e37bbcb05d0a467efb511.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ba89e83329983cfadbfcdda151aaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d42170c7d4249f6b390823606c18c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ed18a92338c7578c18a5ba3a2ae1ed4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/23/d18d825f-ee8b-41df-92c7-06076370bd37.png?resizew=182)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7239b3f2d88c2e45e17e5de9ae1a332.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe6d728b430549f00bb9c0a7bf8bf7d.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8962b0c33df0302e741501c1491bf643.png)
您最近一年使用:0次
2023-10-04更新
|
108次组卷
|
5卷引用:人教A版(2019) 选修第一册 数学奇书 第一章 学业评价(三)
人教A版(2019) 选修第一册 数学奇书 第一章 学业评价(三)人教A版(2019) 选修第一册 数学奇书 第一章 空间向量与立体几何 1.2 空间向量基本定理湖北省襄阳市宜城市第一中学2023-2024学年高二上学期9月月考数学试题(已下线)高二上学期第一次月考十六大题型归纳(基础篇)-2023-2024学年高二数学举一反三系列(人教A版2019选择性必修第一册)(已下线)专题02 空间向量基本定理4种常见考法归类-【考点通关】2023-2024学年高二数学高频考点与解题策略(人教B版2019选择性必修第一册)