解题方法
1 . 已知
,
分别是
轴和
轴上的单位向量.若
,则
,
,
,所以![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bfcd094166964fd95978201511716d4.png)
_________ .从而得向量数量积的坐标表示公式:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bd01707e8d32f277b83c007a5b15bd7.png)
_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bf2a896afb91ca06b513acd1a73bb83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/821a7c2e810ef18a2ee78f3722f03c8c.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/82f741158f16b5cd71cc1fa552b317c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef860c880cd1da63a161aa8a576ab5d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73ee1daee178ca3c866bd1b23d343ab8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/324e1eeeae3ebc7c73654f1a78ecba2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bfcd094166964fd95978201511716d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bd01707e8d32f277b83c007a5b15bd7.png)
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2 . 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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