名校
解题方法
1 . 如图,在平行六面体
中,
,
,
,
,点P满足
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/10/d7de5e5c-6d65-4483-a9f8-d1d4bea1be68.png?resizew=180)
(1)证明:O,P,
三点共线;
(2)求直线
与平面PAB所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a23f01af749100e1888bba06268843db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27db558e8db4c957654c8e5cecd2d2dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d927585a17c2e98ef7d5a9589a26ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/051166055185ba78ed6c4260c14059a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b555561fe89500911f265677801811fd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/10/d7de5e5c-6d65-4483-a9f8-d1d4bea1be68.png?resizew=180)
(1)证明:O,P,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
您最近一年使用:0次
名校
解题方法
2 . 三阶行列式是解决复杂代数运算的算法,其运算法则如下:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/281c685c6a79c8293c8b5083c3a8dc4c.png)
若
,则称
为空间向量
与
的叉乘,其中
,
,
为单位正交基底. 以
为坐标原点、分别以
,
,
的方向为
轴、
轴、
轴的正方向建立空间直角坐标系,已知
,
是空间直角坐标系中异于
的不同两点
(1)①若
,
,求
;
②证明
.
(2)记
的面积为
,证明:
.
(3)证明:
的几何意义表示以
为底面、
为高的三棱锥体积的
倍.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/281c685c6a79c8293c8b5083c3a8dc4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e91aaddb8691f8afa477a96bf630631.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aba64ae92194bc4f0f6e49725471542.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/8643f24c3af715421ec0ccd3224ed453.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d541143135cb9b8166bc631a85ac6a41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a471332d4f3731d90f62fdf819f39824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e73db31aecdde14e0002f082d9091df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2980a18e4d0a2a795b7983a1a1866db7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1821c677712026f8de34fe924b1f52a.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(1)①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41ef077626c88964805a45849471a1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bb22d1c614d99e2639864e43f4b6277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00db2bada2cfc90c5213aca8af17df4c.png)
②证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb8623a42db5ceb745a16d72739f513.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29aa828f2bd9a5e63ee58dcaa9d0d336.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0505ce82dd5726c22fcaac54d01d630b.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8191a760981f2d67648905665c8b167a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad58b362528b814739ceb7fe5febfc28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
您最近一年使用:0次
2024-03-07更新
|
895次组卷
|
8卷引用:河南省部分重点高中2024届高三普通高等学校招生全国统一考试(期末联考)数学试卷
河南省部分重点高中2024届高三普通高等学校招生全国统一考试(期末联考)数学试卷江苏省扬州市仪征中学2024届高三下学期期初调研测试数学试题 河南省部分重点高中(青桐鸣)2023-2024学年高三上学期期末大联考数学试题(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)江苏省江都中学2023-2024学年高二下学期3月联考数学试卷江苏省盱眙中学2023-2024学年高二下学期第一次学情调研数学试题(已下线)第七章 应用空间向量解立体几何问题拓展 专题二 平面法向量求法及其应用 微点2 平面法向量求法及其应用(二)【培优版】(已下线)专题08 期末必刷解答题专题训练的7种常考题型归类-期末真题分类汇编(北师大版2019必修第二册)
名校
解题方法
3 . 如图所示棱长为1的正四面体
,
、
分别为
、
中点,
为靠近
的三等分点.记
,
.
,
,求
的最小值;
(2)求证:
平面
.
![](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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d458bed4c0f3e91667eb8705c9c90d99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/415c6484536cc61efd5529fcb0b15eb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037d31ba398cccf9770084cfdedaf045.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c51159984b2cb00f30b3986315019623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9007ec6447a9428dadb4fdb7ae9bae.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/063510e3c1fb6a7ccc3b8e3e3c7d660e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24dc8826770249f3996b8a188c03da92.png)
您最近一年使用:0次
2024-05-03更新
|
318次组卷
|
3卷引用:浙江省北斗联盟2023-2024学年高一下学期4月期中联考数学试题
浙江省北斗联盟2023-2024学年高一下学期4月期中联考数学试题(已下线)专题06 立体几何初步解答题热点题型-《期末真题分类汇编》(江苏专用)湖北省荆州市荆州中学2023-2024学年高一下学期5月月考数学试卷
名校
4 . 已知三棱柱
,
,
,
为线段
上的点,且满足
.
平面
;
(2)求证:
;
(3)设平面
平面
,已知二面角
的正弦值为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c3e9ef3e849788645552cfb0735d987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/228801f049197a3e30debf8f38fa32ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b276effd97748e08a12333bf3dd78fe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e176de3cd728b538109e9a4fb07e120.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7592c4f01c8e06c7ee90df5b9413a9f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/623f49f1a30f13a3f6706142ed0f92f4.png)
(3)设平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c5c810221d9f1482284486c194c1d38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddd6f80993ce27b2619335e0d83bec57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2830f4565e65043037ce75520530bbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827ccf0c04aa941ba20d5f4c6068b46b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
您最近一年使用:0次
2023-11-08更新
|
1590次组卷
|
4卷引用:江苏省常州市教育学会2023-2024学年高三上学期期中数学试题
江苏省常州市教育学会2023-2024学年高三上学期期中数学试题江西省宜春市宜丰县宜丰中学2024届高三上学期12月月考数学试题(已下线)专题15 立体几何解答题全归类(练习)(已下线)模块三 专题2 解答题分类练 专题3 空间向量线性运算(苏教版)
2024高三·全国·专题练习
解题方法
5 . 如图,在正方体
中,
,
,
,点M,N分别是
,
的中点.
(1)试用
,
,
表示
.
(2)求证:
平面
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a08f3a88dffed011df93d1d606a08ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f8173bb7787b6b107acfe767dd1d91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/232311b4261c36b659555a07bfa00f7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cfbc0b5a8fbde804bd8425a4b76d207.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/3/d1ed7a47-813c-483a-a098-5d06cabb43a5.png?resizew=171)
(1)试用
![](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/4a014dff8997c661055229de29c61cfc.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7592c4f01c8e06c7ee90df5b9413a9f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
您最近一年使用:0次
解题方法
6 . 如图,在底面
为菱形的平行六面体
中,
分别在棱
上,且
,且
.
(1)求证:
共面;
(2)当
为何值时,
;
(3)若
,且
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8e86e3991200297ad172455e5ea93f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5cdcc0d2cbcf7ebf6975618f3114d51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba9196864f83e290af5ba64c4eb2c7ef.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/24/ef05bb8c-102a-4217-a2e1-1184e895085c.png?resizew=160)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b488a540540a2d6a4e3b8b5f67b04611.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f6df10d0b03d6f6e640d9c5f3695a4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1515a445310d259a080d02e16c2e58e.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af0e44cb429eea46e7ee4320147192b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccffde150c595f4e4d444f251c87b1d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
您最近一年使用:0次
2024-04-07更新
|
327次组卷
|
2卷引用:江苏省常州市联盟学校2023-2024学年高二下学期3月阶段调研数学试题
7 . 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
8 . 如图,在正四棱柱
中,
,
,
、
分别为
和
的中点.
;
(2)求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d927585a17c2e98ef7d5a9589a26ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/394c5d2f55221975503be8aa18022480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1384ffba86ff08ce9e783d5d1bc51686.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bff3ccea5989c60e51e321af3f53f54.png)
您最近一年使用:0次
2024-01-27更新
|
224次组卷
|
5卷引用:江苏省南通市2023-2024学年高二上学期期末数学考试
江苏省南通市2023-2024学年高二上学期期末数学考试湖南省株洲市第二中学2021-2022学年高二上学期第三次月考数学试卷江苏省睢宁高级中学2023-2024学年高二下学期3月学情检测数学试卷(已下线)模块一 专题6 《空间向量应用》(苏教版)(已下线)模块三 专题2 解答题分类练 专题3 空间向量线性运算(苏教版)
名校
9 . 如图,在直四棱柱
中,
,
,
,E,F,G分别为棱
,
,
的中点,建立如图所示的空间直角坐标系.
(1)求
的值;
(2)证明:C,E,F,G四点共面.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a11029ca6b4b9e7f777af0280cf163c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a114031e9fd808124cf218d82d5cdc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/394c5d2f55221975503be8aa18022480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/27/88706f8c-b39d-41f9-a3ae-4ff60a08d07d.png?resizew=260)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cf2922c94338fb5c91b8c1ff9bb7e34.png)
(2)证明:C,E,F,G四点共面.
您最近一年使用:0次
2023-11-26更新
|
440次组卷
|
3卷引用:陕西省咸阳市礼泉县2023-2024学年高二上学期期中学科素养调研数学试题
陕西省咸阳市礼泉县2023-2024学年高二上学期期中学科素养调研数学试题(已下线)第6章 空间向量与立体几何单元综合测试卷-【帮课堂】2023-2024学年高二数学同步学与练(苏教版2019选择性必修第二册)江苏省扬州市广陵区红桥高级中学2023-2024学年高二下学期3月月考数学试题
解题方法
10 . 已知空间四点
,
,
和
,求证:四边形
是梯形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356585c75a5db4754720dcab6a58fb50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dc6169910db42dbbd215fadbe90ff67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6ddc9c616b1912bfdfc52e564bf5354.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/164d7860e4b8e67e07fb1e189f984b2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
您最近一年使用:0次
2023-10-05更新
|
274次组卷
|
6卷引用:湘教版(2019)选择性必修第二册课本例题2.3.2空间向量运算的坐标表示
湘教版(2019)选择性必修第二册课本例题2.3.2空间向量运算的坐标表示6.3.4平面向量数乘运算的坐标表示练习(已下线)6.3.2+6.3.3+6.3.4平面向量的正交分解及坐标表示【第三课】“上好三节课,做好三套题“高中数学素养晋级之路(已下线)专题1.4 平面向量基本定理及坐标表示-重难点突破及混淆易错规避(人教A版2019必修第二册)(已下线)模块一 专题2 平面向量基本定理与坐标运算(讲)(已下线)模块一 专题4 平面向量基本定理与坐标运算(讲)北师大版高一期中