名校
解题方法
1 . 已知四面体
的各棱长均为1,D是棱OA的中点,E是棱AB的中点,设
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/19/87ffb349-9c4f-45be-9c94-53d583b6b0cd.png?resizew=187)
(1)用向量
、
、
表示
、
;
(2)判断
与
是否垂直;
(3)求异面直线BD与CE所成的角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4278c0911e7df78965e78cff69cac5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211429ce394297f664f6b8e16edec714.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff2809b448b246cfd11bba7a003c8311.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/19/87ffb349-9c4f-45be-9c94-53d583b6b0cd.png?resizew=187)
(1)用向量
![](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/4781e3daa2c4e018ca0ae09bb56abc0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0735144f6e24b6b32028ff14c17c1cec.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4781e3daa2c4e018ca0ae09bb56abc0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0735144f6e24b6b32028ff14c17c1cec.png)
(3)求异面直线BD与CE所成的角.
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名校
2 . 已知四棱锥
的底面是平行四边形,平面
与直线
,
,
分别交于点
,
,
且
,点
在直线
上,
为
的中点,且直线
平面
.
![](https://img.xkw.com/dksih/QBM/2020/11/25/2600441359441920/2602225913307136/STEM/61815a03ddba417ca83b5f229c413aa0.png?resizew=265)
(1)设
,
,
,试用基底
表示向量
;
(2)证明,四面体
中至少存在一个顶点从其出发的三条棱能够组成一个三角形;
(3)证明,对所有满足条件的平面
,点
都落在某一条长为
的线段上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28121a595e617a54a3432bf5119b8773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e98c4b3f3fe826e124ca7d199d4ca4b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c59ab3c430815c8e1a5cef009876e6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/558432772e71c0909a2764efbecaccf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/817a419430d9951cbdb89b657b21bcf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7592c4f01c8e06c7ee90df5b9413a9f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://img.xkw.com/dksih/QBM/2020/11/25/2600441359441920/2602225913307136/STEM/61815a03ddba417ca83b5f229c413aa0.png?resizew=265)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74d86e0dea2a956b5db60e6ae6632517.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/265f935c4ac4f0659c2d6ee01a5ae8f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df27380eba37f02650e85ae6ec751d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8d2051594370095e72e173fd95888a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8434200b36130e0fdf8f0b673a3bb09.png)
(2)证明,四面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1198837efde80d1b090e2358e958f397.png)
(3)证明,对所有满足条件的平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41f6345f54cdba8572baeb130df483b7.png)
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2020-11-27更新
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13卷引用:3.1空间向量及其运算(作业)(夯实基础+能力提升)-【教材配套课件+作业】2022-2023学年高二数学精品教学课件(沪教版2020选择性必修第一册)
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解题方法
3 . 如图,已知
,
为异面直线,
,
在
上,
,
在
上,
,
,
,
.求
,
所成的角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.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/0a6936d370d6a238a608ca56f87198de.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/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aed2668385c8f295ecc62ff721267d1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/208b71ecd0d9594374170fa8d6a32bd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037b342a682cbd4241855a243da3c016.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://img.xkw.com/dksih/QBM/2020/6/25/2492402308710400/2492996224147456/STEM/49b53958a4f14ae58d27f885d95a04cd.png?resizew=168)
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