解题方法
1 . 如图,在底面为矩形的四棱锥
中,
底面
,
为棱
上一点,且
,
以
为坐标原点,
的方向为
轴的正方向,建立如图所示的空间直角坐标系.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/30/12783d5c-ab43-4ecb-a9db-6e6e927a1699.png?resizew=173)
(1)写出
,
,
,
四点的坐标![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32ca6fa9955690cec01db601e3abce0c.png)
(2)求
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db807b09cc550f476b3f8fa0c6a14425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30b0393ce62b24aa5f9b740d4cc6743b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8715f8038c86e138e9a7f57868f1299c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e773bfc45764fd279a64990c98ccd61e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/30/12783d5c-ab43-4ecb-a9db-6e6e927a1699.png?resizew=173)
(1)写出
![](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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32ca6fa9955690cec01db601e3abce0c.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fa7e366a255f5b3f45e4d7c0cf34486.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/894181d9611539ed340e278adaaf8ab1.png)
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名校
解题方法
2 . 在正方体
中,
分别是
和
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/29/bba315d7-e3a6-4b2c-b22e-7e4d3667e029.png?resizew=169)
(1)求异面直线
和
所成角的大小;
(2)求证:平面
平面
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd4cf99a0d5833beacc3a0ee39d39458.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b48386aac017c768709a87cda4b7001b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a82f09a3515f297f0edd47c24718ff9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/29/bba315d7-e3a6-4b2c-b22e-7e4d3667e029.png?resizew=169)
(1)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4821e902e3ecd5d604a5827361a9344.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c52091eb745de866044477641a7c55f.png)
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3 . 如图,线段PC、BC、DC两两垂直,AD∥BC,CB=CD=CP=3AD=3.点F为PA的中点,点E在CD上,且CE=1.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/30/c8f796cc-d1a3-48b0-801b-53cdadca3b88.png?resizew=207)
(1)求证:BE⊥CF;
(2)求平面ADP与平面BPC夹角的余弦值.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/30/c8f796cc-d1a3-48b0-801b-53cdadca3b88.png?resizew=207)
(1)求证:BE⊥CF;
(2)求平面ADP与平面BPC夹角的余弦值.
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5卷引用:云南省昆明市官渡区艺卓中学2022-2023学年高二上学期9月月考数学试题
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4 . 在边长为3的正三角形ABC中,E,F,P分别是AB,AC,BC边上的点,且满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/682ff07b43d18fac5a4bb6431218114a.png)
(如图1),将
沿EF折起到
的位置,使二面角
成直二面角,连接
,
(如图2).
;
(2)求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/682ff07b43d18fac5a4bb6431218114a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29ae300e4ff3e89d0e1e5671eacd9585.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb2d555062f34d5a74f6d47da4ea8888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc14ed237a4bcc35cbd1f5f1321b3718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26d9636ad77369535852c6e4493446a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0ff310aabd2282b539537ebed3f788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad55104d5104c134ddb44cee5a8bfaba.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8097ceec369f4de5071a58290ce6e7e9.png)
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5 . 如图,在四棱锥
中,底面ABCD为矩形,
平面PAD,E是AD的中点,
为等腰直角三角形,
,
.
;
(2)求PC与平面PBE所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f9157fce2a8339d281178c7c0bccbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55a675310c8ba418e5a59beb7317e21e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1676715fa1188b716cc945be7b94e13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af62a8c94bdc27efa2ec03e58d9400ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d31767eb718a0327eca546fe6a189cb.png)
(2)求PC与平面PBE所成角的正弦值.
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8卷引用:河南省豫北名校2022-2023学年高二上学期9月教学质量检测数学试题
河南省豫北名校2022-2023学年高二上学期9月教学质量检测数学试题湖南省益阳市2022-2023学年高二上学期10月联考数学试题河南省郑州市新密市第一高级中学2022-2023学年高二上学期10月月考数学试题河南省平顶山市叶县高级中学2023-2024学年高二上学期10月月考数学试题广东省深圳市深圳外国语学校2023-2024学年高二上学期10月月考数学试题山西省朔州市怀仁市第一中学校2023-2024学年高二上学期第三次月考(11月)数学试题河南省新乡市获嘉县第一中学2023-2024学年高二上学期第一次月考数学试题吉林省四平市第一高级中学2023-2024学年高二上学期期初验收考试数学试题
名校
6 . 如图甲,在矩形
中,
为线段
的中点,
沿直线
折起,使得
,如图乙.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/3/f95147e6-220a-4bb2-ba0f-0e66623a6b32.png?resizew=368)
(1)求证:
平面
;
(2)线段
上是否存在一点
,使得平面
与平面
所成的角为
?若不存在,说明理由;若存在,求出
点的位置.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/438f5e0b08422b2f93d122341b3d7672.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c8110f066b294763b30456f7cd90d1e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/3/f95147e6-220a-4bb2-ba0f-0e66623a6b32.png?resizew=368)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662698361c6b3ddaf0c28a3c87be53e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
(2)线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/977a483bd52c08968f4097d10609be20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
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15卷引用:广东省四中、三中、培正三校2022-2023学年高二上学期期中联考数学试题
广东省四中、三中、培正三校2022-2023学年高二上学期期中联考数学试题广东省华南师范大学附属中学2022-2023学年高二上学期期中数学试题辽宁省大连市第二十四中学2022-2023学年高二上学期期中数学试题广东省深圳市福田区红岭中学2022-2023学年高二上学期期中数学试题辽宁省大连市第三十六中学2022-2023学年高二上学期期中数学试题浙江省金华市2022-2023学年高二上学期期末数学试题广东实验中学附属江门学校2022-2023学年高二上学期期中数学试题(普高班)广东省东莞实验中学2022-2023学年高二上学期第二次月考数学试题重庆市第八中学校2023届高三上学期高考适应性月考(一)数学试题广东省梅州市五华县水寨中学2022-2023学年高三上学期10月月考数学试题福建省厦门海沧实验中学2023-2024学年高二上学期10月阶段性检测数学试题广东省汕头市金山中学2023-2024学年高二上学期期中数学试题广东省东莞市四校2023-2024学年高二上学期期中联考数学试题四川省成都市列五中学2023-2024学年高二上学期12月月考数学试题四川省绵阳市南山中学2022-2023学年高三下学期3月月考数学(理)试题
名校
解题方法
7 . 如图,在四棱锥
中,
平面
,底面
为矩形,
,G为
的重心,M为线段
的中点,
与
交于点F.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/28/aeab79dc-6152-4bee-8032-4837ca38d0c7.png?resizew=183)
(1)当
时,证明:
平面
;
(2)当平面
与平面
所成锐二面角为
时,求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5164a3cc47e266446d49127e2ef10c37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2d2a8070f1a70c76686847697146383.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/28/aeab79dc-6152-4bee-8032-4837ca38d0c7.png?resizew=183)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70dc2c20619a4fc12a0cfda59af5b69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36148e5b0d89ba45bd98b91da00bf2b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
(2)当平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac3bbe7410b0176a1b3f9410ab761be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c69f7f73a7f0e0b5a0a82a51f8ab28.png)
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5卷引用:河南省中原名校2022-2023学年高二上学期第一次联考数学试题
河南省中原名校2022-2023学年高二上学期第一次联考数学试题广西柳州市第三中学2022-2023学年高二上学期10月月考数学试题河南省夏邑县会亭高级中学2022-2023学年高二上学期第一次月考数学试题高二数学试题-中原名校2022-2023学年高二上学期第一次联考试题(已下线)第二章 立体几何中的计算 专题四 空间体积的计算 微点1 空间图形体积的计算方法【基础版】
名校
8 . 如图,在直三棱柱
中,
,E为线段
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/dbb189e6-c49c-4401-8f34-3e87ff8e4e92.png?resizew=164)
(1)证明:
平面
;
(2)若
,求二面角
的平面角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1b622af0a12049cefde78d28d9a2597.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/dbb189e6-c49c-4401-8f34-3e87ff8e4e92.png?resizew=164)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4cc24037ff3b29f2cb81291734869d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db3ef97d64e58d311019b70fe5e2cc0d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad1a56baf43ffdf67bc8460856e31fec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02b20ef9d660baa101c0574c46e107e0.png)
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7卷引用:河南省中原名校2022-2023学年高二上学期第一次联考数学试题
名校
解题方法
9 . 如图所示,在直三棱柱
中,
,点
是
的中点,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/27/714fb243-a560-44a5-a038-9aaec3047771.png?resizew=154)
(1)求异面直线
与
所成角的余弦值;
(2)求直线
与平面
所成的角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a3cc9cccfb4c260dac05f4ed57e8c10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c04b0b1fd6979d5cf1d7be8f5109186a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f32a7f0fb3cc6eaa91ad3f44b9d5610.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/27/714fb243-a560-44a5-a038-9aaec3047771.png?resizew=154)
(1)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabea664e61863b3b3279dbce607924e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fd4c85bb98a2a0afddd7ed92578ad2e.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabea664e61863b3b3279dbce607924e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
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4卷引用:山西省浑源县第七中学校2022-2023学年高二上学期第一次阶段性学情检测数学试题
山西省浑源县第七中学校2022-2023学年高二上学期第一次阶段性学情检测数学试题黑龙江省哈尔滨市剑桥第三中学2022-2023学年高二上学期9月月考数学试题辽宁省朝阳市建平县实验中学2022-2023学年高二上学期第二次月考数学试题(已下线)第06讲 向量法求空间角(含探索性问题) (高频考点—精讲)-1
2023高三·全国·专题练习
名校
解题方法
10 . 如图,正四面体ABCD(所有棱长均相等)的棱长为1,E,F,G,H分别是正四面体ABCD中各棱的中点,设
,
,
,试采用向量法解决下列问题:
(1)求
的模长;
(2)求
,
的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e7a93a1399ff7a2bde342652479241b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae92f0c583cc9daf980a8621ad96aef5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89f111250ea56c59b179cfc7b5db12cd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/12/2ae55861-e7c0-4a16-8bac-8e9b6fec1050.png?resizew=165)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6b5a51e65e9b56a9970de33ead263a9.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6b5a51e65e9b56a9970de33ead263a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21162e18e3c389215ac1a44858d6c5a7.png)
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2022-09-26更新
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4卷引用:广西玉林市第十一中学等校2023届高二上学期期中联合测试数学试题
广西玉林市第十一中学等校2023届高二上学期期中联合测试数学试题河南省郑州市回民高级中学2022-2023学年高二上学期期中数学试题(已下线)第02讲 1.1.2空间向量的数量积运算(7类热点题型讲练)-【帮课堂】2023-2024学年高二数学同步学与练(人教A版2019选择性必修第一册)(已下线)第51讲 空间向量的概念