名校
1 . 已知正方体
的棱长为1,点P满足
,
,
,
(P,B,D,
四点不重合),则下列说法正确的是( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ae7072624587654d162548a80d7a1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be7b5066a7ac79c102d2a30d6280d3ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
A.当![]() ![]() |
B.当![]() ![]() ![]() ![]() |
C.当![]() ![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() ![]() |
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2023-12-09更新
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808次组卷
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8卷引用:广东省执信、深外、育才等学校2024届高三上学期12月联考数学试题
名校
2 . 如图所示的几何体
中,
和
均为以
为直角顶点的等腰直角三角形,
,
,
,
,
为
的中点.
![](https://img.xkw.com/dksih/QBM/2020/5/26/2471193277857792/2471708998156288/STEM/671926a565504cf289445e9b734199ea.png?resizew=237)
(1)求证:
;
(2)求二面角
的大小;
(3)设
为线段
上的动点,使得平面
平面
,求线段
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be89b9d1709d7974a108142c5fa2ccec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a855335176fc36a15017f50a8561348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/099dd87a526391f830ac2a11e7d7ad56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7bd02e0adeae92ba9526261b1baf797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52bbb16f9380dd62d556480a3944be31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66418ef39d3081d89411a4907d8599f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fdf241efcd0c8026d188fad5a5ba4e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://img.xkw.com/dksih/QBM/2020/5/26/2471193277857792/2471708998156288/STEM/671926a565504cf289445e9b734199ea.png?resizew=237)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f5ffe436f8eb53a211abf95baed8ca9.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd548dfcef01e147e4dce25bd384f9b9.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df0cf60b84dcb4baf97c39fe659e08a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910043595f4eed0c5b2a2246bec3664c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
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2020-05-27更新
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2400次组卷
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16卷引用:山东省潍坊市寿光市现代中学2020-2021学年高二(上)期中数学试题
山东省潍坊市寿光市现代中学2020-2021学年高二(上)期中数学试题辽宁省大连市红旗高级中学2020-2021学年高二上学期期中数学试题广东省深圳市福田区外国语高级中学2023-2024学年高二上学期期中数学试题2020届天津市河西区高考一模数学试题山东省新泰市第一中学老校区(新泰中学)2020-2021学年高二上学期第一次月考数学试题山东省寿光现代中学2020-2021学年高二11月月考数学试题(已下线)考点29 空间向量解决空间直线、平面位置关系-备战2021年新高考数学一轮复习考点一遍过(已下线)专题1.3 空间向量及其运算的坐标表示-2020-2021学年高二数学同步课堂帮帮帮(人教A版2019选择性必修第一册)海南省北京师范大学万宁附属中学2021-2022学年高二上学期第一次月考数学试题天津市南开中学2022-2023学年高三上学期10月阶段性统一练习(一)数学试题广西玉林市博白县第四中学(博白县中学书香校区)2022-2023学年上学期高二9月月考数学试题广东省台山市第一中学2022-2023学年高二上学期期末数学试题(已下线)第01讲 空间向量及其运算(6大考点)-2022-2023学年高二数学考试满分全攻略(人教A版2019选择性必修第一册)(已下线)专题01 空间向量与立体几何(5)(已下线)高二数学第一学期期期末押题密卷03卷(已下线)第3章 空间向量及其应用 单元综合检测(难点)-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)
3 . 已知
,
,
和
为空间中的4个单位向量,且
,则
不可能等于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/969604545902c9a66549a4a44ec3a3c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90ab17fd4247cdd710c363d5d3fbc5bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8c308ea87b699ee1dcb879a568899de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/358d896d9eab84c6695ddc25196e6857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e7b8925a75bc1ef601edb37dbfb9f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f59ca8f14b92276cf540b7a0166c63ae.png)
A.3 | B.![]() | C.4 | D.![]() |
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2019-01-03更新
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3257次组卷
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7卷引用:上海市复兴高级中学2022届高三上学期期中数学试题
上海市复兴高级中学2022届高三上学期期中数学试题四川省雅安市天立学校2022-2023学年高二下学期期中教学质量测试数学(理)试题【校级联考】浙江省金丽衢十二校2019届高三第二次联考数学试题【校级联考】浙江省丽水市四校联考2018-2019学年高二5月阶段性考试数学试题(已下线)1.1 空间向量及其运算-2021-2022学年高二数学尖子生同步培优题典(人教A版2019选择性必修第一册)上海市黄浦区大同中学2022届高三上学期12月月考数学试题(已下线)专题01 空间向量及其运算压轴题(5类题型+过关检测)-【常考压轴题】2023-2024学年高二数学上学期压轴题攻略(人教A版2019选择性必修第一册)
名校
解题方法
4 . 点P是长方体
内的动点,已知
,Q是平面BC₁D上的动点,满足
,则
的最小值是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aecba711ddb1b9181652eceb18fb038b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4d14c6959273338a048b023805cce80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc5604d3e156df3e7ccca0ccec9c9d45.png)
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2023-11-11更新
|
429次组卷
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5卷引用:浙江省浙南名校联盟2023-2024学年高二上学期11月期中联考数学试题
浙江省浙南名校联盟2023-2024学年高二上学期11月期中联考数学试题浙江省浙南名校联盟2023-2024学年高二上学期期中联考数学试题浙江省金华市武义第一中学2023-2024学年高二上学期11月检测2数学试题(已下线)3.3 空间向量的坐标表示(九大题型)(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)压轴小题5 空间向量中的最值问题
名校
5 . 正方体
中,点P满足
,且
,直线
与平面
所成角为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecbbb611974a7dcc236fd1fab7fc4c36.png)
_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c3c9fd40623dadf10a50c77caa214fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd24c686fbaaa68705d654b880481ffe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6655c413f509068d30b165f9d92bdba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da7977ab975efa6411cc17de39be70d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecbbb611974a7dcc236fd1fab7fc4c36.png)
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6 . 若
,则称
为
维空间向量集,
为零向量,对于
,任意
,定义:
①数乘运算:
;
②加法运算:
;
③数量积运算:
;
④向量的模:
,
对于
中一组向量
,若存在一组不同时为零的实数
使得
,则称这组向量线性相关,否则称为线性无关,
(1)对于
,判断下列各组向量是否线性相关:
①
;
②
;
(2)已知
线性无关,试判断
是否线性相关,并说明理由;
(3)证明:对于
中的任意两个元素
,均有
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94d935457799f234e86a59e2f662d5ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00d8f90655e6341907aa9c7c62d4398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/554f144d15fc567b25935b38917430c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aff8d9b6533ff319420cdc5e8740b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e1156e087d0c92bf15ea7a53d021fcc.png)
①数乘运算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ee3d2d5f39d4050799537d5ad6bb375.png)
②加法运算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/374036b2d97be3bb771c6d1bfd2ae6eb.png)
③数量积运算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3760d1b114cab294d2b8af405de49814.png)
④向量的模:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1e7f7030526d65d5fff785d0d35a6ba.png)
对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00d8f90655e6341907aa9c7c62d4398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab882b48940b6e1a185a513a0d8e8d97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f3e3f607c7d170c8a9e614bfd2cb5f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/946f81ae87f9c8cdc1017af6c1ec2fb2.png)
(1)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b56eb5f6747ea94d1075210265214211.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee6432103414cd0efd40a0c0017eb11b.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bf1bee58a691b31e06e088afed4c25c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a76eb5fb9c09b830eecb5ba7efea4e09.png)
(3)证明:对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00d8f90655e6341907aa9c7c62d4398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c21920a0a39b1604e130601f061b056.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1120f6ff618146c9851df5dea05c1f55.png)
您最近一年使用:0次
名校
7 . 如图,三棱锥
中,平面
平面
,
,
,
,
,
,点
是棱
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/1/97233532-de88-432e-9680-0ca14b88702d.png?resizew=176)
(1)求证:
平面
;
(2)求直线
与平面
所成角的正弦值;
(3)设点
是线段
的中点,棱
上是否存在点
,使得
平面
?若存在,求
的值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d077f6da8b2c00b152d4679aa2ed7f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/036de574712cad14bddadf6653c7e714.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cb96e0331eebe80ed1ff610faf531fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/615fc8790237a1b09af51d6bcad6b595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb5b12692517a39c320f99a479eb055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7aeb2a8d1437eeb4482c3b6ad9f315.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/1/97233532-de88-432e-9680-0ca14b88702d.png?resizew=176)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2730b513bd3359c3dfe6567e04f5ef9.png)
(3)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a46435220a682a6f67d7ac8608be1c7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92d53cf3ab15beaf9960569932ad0812.png)
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