向量的数量积
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/15/40ee25a3-7b2a-4555-9ede-deee420bc0b9.png?resizew=158)
(1)向量数量积的定义
①向量的夹角:已知两个非零向量
,
,O是平面上的任意一点,作
=
,
=
(如图所示),则∠AOB=θ(0≤θ≤π)叫做向量
与
的夹角.
②向量的平行与垂直:当θ=0时,
与
同向;当θ=π时,
与
反向;如果
与
的夹角是
,我们说
与
垂直,记作
⊥
.
③向量的数量积:已知两个非零向量
与
,它们的夹角为θ,我们把数量|
||
|cosθ叫做向量
与
的数量积(或内积),记作
·
,即
·
=|
||
|cosθ.
规定:零向量与任一向量的数量积为0.
(2)向量的投影
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/15/7d24cdee-e446-434b-9a38-a362bc57a47f.png?resizew=165)
①定义:如图,设
,
是两个非零向量,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf6eab0544c9fad5acde25bec349fb02.png)
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad9d86315677a86bb4dc09d06310f3f1.png)
,作如下的变换:过
的起点
和终点
,分别作
所在直线的垂线,垂足分别为
,
,得到
,则称上述变换为向量
向向量
投影,
叫做向量
在向量
上的投影向量.
②计算:设与
方向相同的单位向量为
,
与
的夹角为θ,则向量
在向量
上的投影向量是|
|cosθ
.
(3)向量数量积的性质
设
,
是非零向量,它们的夹角是θ,
是与
方向相同的单位向量,则
①
·
=
·
=|
|cosθ.
②
⊥
⇔
·
=0.
③当
与
同向时,
·
=|
||
|;当
与
反向时,
·
=-|
||
|.特别地,
·
=|
|2或|
|=
.
④|
·
|≤|
||
|.
(4)向量数量积运算的运算律对于向量
,
,
和实数λ,有
①
·
=
·
;
②(λ
)·
=λ(
·
)=
·(λ
);
③(
+
)·
=
·
+
·
.
(5)数量积的坐标表示
设
=(x1,y1),
=(x2,y2),则
①
·
=x1x2+y1y2;
2=
;![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dba7843f0a56f5e731c14a4f1edde55f.png)
____________ .
②
⊥
⇔____________ .
③
④设θ是
与
的夹角,则cosθ=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/421d881ac25b59201f971aafba71ffb8.png)
____________ .
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/15/40ee25a3-7b2a-4555-9ede-deee420bc0b9.png?resizew=158)
(1)向量数量积的定义
①向量的夹角:已知两个非零向量
![](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/a4f605ec0729ce6d72237ad662a06862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc9656d8286c4d6fa309d6ae347c89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
②向量的平行与垂直:当θ=0时,
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](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/b1ad72d7565699d1ebb741eb0ce12bac.png)
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
③向量的数量积:已知两个非零向量
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
规定:零向量与任一向量的数量积为0.
(2)向量的投影
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/15/7d24cdee-e446-434b-9a38-a362bc57a47f.png?resizew=165)
①定义:如图,设
![](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/bf6eab0544c9fad5acde25bec349fb02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad9d86315677a86bb4dc09d06310f3f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.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/bc1070a28cb9cb8553c29747d1993b16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65b764e1eb495a092699a2008a031335.png)
![](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/65b764e1eb495a092699a2008a031335.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
②计算:设与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
(3)向量数量积的性质
设
![](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/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
②
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
③当
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4545c2c12384f72614ad95b486a2ace.png)
④|
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(4)向量数量积运算的运算律对于向量
![](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/071a7e733d466949ac935b4b8ee8d183.png)
①
![](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/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
②(λ
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
③(
![](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/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
(5)数量积的坐标表示
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
①
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fd4492bf478b5027e578641fbe74170.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dba7843f0a56f5e731c14a4f1edde55f.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3fc8f299a6555155891a48d89963fbe.png)
④设θ是
![](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/421d881ac25b59201f971aafba71ffb8.png)
2022高一·全国·专题练习 查看更多[1]
更新时间:2022-12-06 16:58:38
|
相似题推荐
填空题-单空题
|
容易
(0.94)
名校
解题方法
【推荐1】已知向量
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac6f62dcd1ebd05a5b351a5ef719c574.png)
___________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102bfda2b5aea3dfe44b8efd474f6b59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4078e7fd4c876a09c7de6157ef78b63d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac6f62dcd1ebd05a5b351a5ef719c574.png)
您最近一年使用:0次
填空题-单空题
|
容易
(0.94)
解题方法
【推荐2】已知向量
,
,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc009cdbdb648e63e75034b9763e1e79.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2c3532ceef53f44152eebf2da34e86e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74789e53eb8f46253b56b8b4c4567f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa5aa846a5b7c96fe2ce665eb1ea5f0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc009cdbdb648e63e75034b9763e1e79.png)
您最近一年使用:0次
填空题-概念填空
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容易
(0.94)
【推荐1】平面向量数量积的坐标表示
设非零向量
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9afc7bfbc67ff08ae5380508d9ca746.png)
_____________ .这就是说,两个向量的数量积等于它们______________ .
平面向量模的坐标形式
(1)若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda3382c51e260b025dc8c95992a78d4.png)
____________ ,或![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c987ded20c4a12204e0bb290de4809.png)
____________ .
(2)如果表示向量
的有向线段的起点和终点的坐标分别为
,那么
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c987ded20c4a12204e0bb290de4809.png)
__________ .
平面向量垂直的充要条件的坐标表示
设
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94d6468ac363b1c2b09ecaf4d0533978.png)
______________ .
平面向量夹角的坐标表示
设
都是非零向量,
,
是
与
的夹角,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47561c477cd43b6f9baba4d32e458082.png)
_________ .
[微思考]若两个非零向量的夹角满足
,则两向量的夹角
一定是钝角吗?
___________
设非零向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c44622a3b1070041dd5786800081501.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9afc7bfbc67ff08ae5380508d9ca746.png)
平面向量模的坐标形式
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fba056a8ce034c16e3570f4331d5b14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda3382c51e260b025dc8c95992a78d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c987ded20c4a12204e0bb290de4809.png)
(2)如果表示向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb80eb942aafb194fadc473776f35b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b6037359d2727b05ee33db9e2c36226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c49cd87ce066439278ffb373ec160c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c987ded20c4a12204e0bb290de4809.png)
平面向量垂直的充要条件的坐标表示
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c44622a3b1070041dd5786800081501.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94d6468ac363b1c2b09ecaf4d0533978.png)
平面向量夹角的坐标表示
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ccba3b87a8a48ac3dd5f72d00bdb1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c44622a3b1070041dd5786800081501.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb80eb942aafb194fadc473776f35b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/433b94c39737727e53468df419d8314a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47561c477cd43b6f9baba4d32e458082.png)
[微思考]若两个非零向量的夹角满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d4e16155c0a82a09e3f6e160df42f63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
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填空题-单空题
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容易
(0.94)
解题方法
【推荐2】已知向量
,
,则它们的夹角![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9251dff989f7d60db751b73033dee269.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a41dfbb644f602fb292a46d57ee3065.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/682038036290d0f64f91244352da25fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9251dff989f7d60db751b73033dee269.png)
您最近一年使用:0次