1 . 夹角
(1)求异面直线所成的角
若两异面直线
所成角为
,它们的方向向量分别为
,则有
=______ .
(2)求直线和平面所成的角
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/19/d419b8ee-ebff-4daf-8cec-3fdeba1d82ae.png?resizew=148)
设直线
的方向向量为
,平面
的法向量为
,直线与平面所成的角为
,
与
的角为
,则有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b363797455e171e5dff9193eb28cc9.png)
______ =_______ .
(3)求二面角
如图,若
于A,
于B,平面PAB交
于E,则________ 为二面角
的平面角,∠AEB+∠APB=180°.若二面角
的平面角的大小为
,其两个面
的法向量分别为
,则
=______ =_______
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/5d787649-3529-4f7f-8a52-6f2716bc421c.png?resizew=224)
(4)求平面与平面的夹角
平面与平面相交,形成四个二面角,把这四个二面角中不大于90°的二面角称为平面
与平面
的夹角![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90e888ea14da893971e13858945bf0cd.png)
_________ =___________ .
(1)求异面直线所成的角
若两异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cc20c2da40acda10e0a4d0477d88d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefd06c239145a2b6ae87a955aa51414.png)
(2)求直线和平面所成的角
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/19/d419b8ee-ebff-4daf-8cec-3fdeba1d82ae.png?resizew=148)
设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de8632651e05bbebcc3e7f6fd0a9d27e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36f1a8e551cba7ec9f451749f60e628d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de8632651e05bbebcc3e7f6fd0a9d27e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36f1a8e551cba7ec9f451749f60e628d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b363797455e171e5dff9193eb28cc9.png)
(3)求二面角
如图,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29e35f3a470885d88519e1a71db4b323.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d4d7c644adf1763379806644b7729c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/754bbd99327195520a4ca3ce3b9a0577.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/754bbd99327195520a4ca3ce3b9a0577.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6656874a99a8251ca5ec10fdab3bee24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9208f329244864318649d828efa6ce2c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/5d787649-3529-4f7f-8a52-6f2716bc421c.png?resizew=224)
(4)求平面与平面的夹角
平面与平面相交,形成四个二面角,把这四个二面角中不大于90°的二面角称为平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90e888ea14da893971e13858945bf0cd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/19/5ed8a498-aa79-418c-8dfb-790d2fb5777f.png?resizew=170)
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2 . 设
分别为两条异面直线
的方向向量,且
,则异面直线
所成的角为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1238a7a8ce9db1a47ae0ac80d5ed70cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22b4218f00da487d3f63b9360144708f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9127f6f433071d510f76b332a3636904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22b4218f00da487d3f63b9360144708f.png)
您最近一年使用:0次
2021-12-23更新
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369次组卷
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2卷引用:湖南省邵阳市第二中学2021-2022学年高一下学期期末数学试题