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解题方法
1 . 定义空间中既有大小又有方向的量为空间向量.起点为
,终点为
的空间向量记作
,其大小称为
的模,记作
等于
两点间的距离.模为零的向量称为零向量,记作
.空间向量的加法、减法以及数乘运算的定义与性质和平面向量一致,如:对任意空间向量
,均有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53bcfdf754e71318fb8329b8e7c09264.png)
,
,
;对任意实数
和空间向量
,均有
;对任意三点
,均有
等.已知体积为
的三棱锥
的底面均为
,在
中,
是
内一点,
.记
.
(1)若
到平面
的距离均为1,求
;
(2)若
是
的重心,且对任意
,均有
.
(i)求
的最大值;
(ii)当
最大时,5个分别由24个实数组成的24元数组
满足对任意
,均有
,且对任意
均有
求证:
不可能对任意
及
均成立.
(参考公式:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c082be7f93f355e1ca70588a4a89aead.png)
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28fde0a8b4ec1e2fff42cee3fc54c0f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28fde0a8b4ec1e2fff42cee3fc54c0f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49da589810153e2ec39ed656a2b61f4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b12de8a4f788ff23d36e74c811354779.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e12e95f703ad30ab9a3d38376830989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53bcfdf754e71318fb8329b8e7c09264.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea3ff5e2f25dfebafaf8db07712ff706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff47a4801df7bc7bce1cb52327a7b174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e0a953946d9e878aa017c7f24ffb40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0714b48d55f6b0854fb90a4255bc49c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6fa157b4f65f3a9aa1f7f82de02e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e1e19465c82977a26ca6900622ee1bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/718ba76bf48024ca425948e470e60042.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c761455094dc4913de76122017a243dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e48ac6b0dda0647d7dad3287ce4ad258.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d131fd570dc36b912396dc2dd06405c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4aeda1e642ce85f1c0394bc419bda8e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f49f84442a1b38f27ac977214cd4b688.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/902a402a179a09f74f2391fb5cb4ae6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/247daad150250fc13a230d5375adda93.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab39849dc21c8c68cd5cde0911d5db23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a61e6011a0717ef57516821d0407a656.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aae3155971b2bb3c9d68b43e14b7186f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ffe9f4e3243bd760835af03fa7ffe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c053ebe33366203ad0eca474760118.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d05f59bfd6b1f55920e73653bf87a46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ffe9f4e3243bd760835af03fa7ffe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db09e9844b90e46a6f2f5a710b6a3451.png)
(参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c082be7f93f355e1ca70588a4a89aead.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2343b61be295955a2b9baea86202f32.png)
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2卷引用:福建省厦门第一中学2023-2024学年高一下学期6月适应性练习数学试卷
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2 . 在四面体OABC中,
是棱OA上靠近
的三等分点,
分别是
的中点,设![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e33a7fe8acce3fc214d44459e92ffa97.png)
,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee477d164d38ff067fc4ce342cf6d380.png)
_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/518407c06c4c966f670ea9d2414984a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fa87ac2b7d6a58f5aaee923d83f799b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e33a7fe8acce3fc214d44459e92ffa97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71af5e80069433636f4ef0227a508dc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a8d49bf33e578b25811e22e26dbf584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee477d164d38ff067fc4ce342cf6d380.png)
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3 . 在三棱锥
中,
和
都是等边三角形,
,D为AB中点,则
的值是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fa9fa88d321bc55ca505771d77dc405.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1605925e08c31d30c193c4321aacba26.png)
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4 . 已知四面体
中,
为
中点,若
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25a2efd3f26a34c6f7b6277e009836ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34b1fe1294cb5b9b8a8762b4a16e920d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5085e3cdef9ea6c564e079f745d6fdb.png)
A.3 | B.2 | C.![]() | D.![]() |
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5 . 在平行六面体
中,
为
延长线上一点,且
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efb69ae7bc9b325bd8d128a78fb2dcaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/837755b76f7baa821e707ac519d05b2b.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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6 . 在棱长为2的正四面体A-BCD中,E,F分别是AD,BC的中点,G是△BCD的重心,则下列结论正确的是( )
A.![]() | B.![]() |
C.![]() ![]() ![]() | D.![]() |
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7 . 如图所示,空间四边形
中,
,点
分别为
上的点,且
为
中点,则
等于( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/22/16637f6c-a65c-4485-b1e8-5ce8b817380c.png?resizew=147)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c4cf572c0cb3e8b3978b5f834bd9b61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfc3ee16ccc10bb5a1ddd181a3f9989b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6623a893148b78d2898baff3a279858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a014dff8997c661055229de29c61cfc.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/22/16637f6c-a65c-4485-b1e8-5ce8b817380c.png?resizew=147)
A.![]() | B.![]() |
C.![]() | D.![]() |
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8 . 如图,在三棱柱
中,M为
的中点,设
,
,
,则
( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/4/f4da81bc-ed3a-4026-93fd-db130c0db4d8.png?resizew=134)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a79f2cfa5e197f859d8cac3d2287ece9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0e72403ac1926f555073d01fa22ab33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddf9d421e75be0122791cd2cd4b1cfdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347a2f174217898b8148515dc29ab948.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/4/f4da81bc-ed3a-4026-93fd-db130c0db4d8.png?resizew=134)
A.![]() | B.![]() | C.![]() | D.![]() |
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5卷引用:福建省福州市长乐第一中学2023-2024学年高二上学期1月月考数学试题
福建省福州市长乐第一中学2023-2024学年高二上学期1月月考数学试题辽宁省辽阳市2023-2024学年高二上学期1月期末考试数学试卷内蒙古2023-2024学年高二上学期期末教学质量检测数学试题(已下线)专题11 空间向量及其运算10种常见考法归类(1)(已下线)第6章 空间向量与立体几何 章末题型归纳总结-【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)
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9 . 如图,平行六面体
的底面是菱形,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ef3d3128ce3f5799c39dd6b8290122e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/1/d4011eb3-e96d-4922-b168-9d7faf9d4842.png?resizew=167)
(1)用空间的一个基底
表示
,并求
的长;
(2)求异面直线
与
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ef3d3128ce3f5799c39dd6b8290122e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/1/d4011eb3-e96d-4922-b168-9d7faf9d4842.png?resizew=167)
(1)用空间的一个基底
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ed66bde4aaae35dc7a1ccc292a6ba9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/053c0f6846f2bf8671b351a4263a0270.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
(2)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabe764f05300ac83c7d16b685d27af4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2eb89294b31ffdd2680b4361e8994d7.png)
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10 . 已知A,B,C,D是空间中互不相同的四个点,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f94481b5b90278632dabf31c07437f8.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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