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1.1.2 矢量场的基本运算
除去矢量除法没有定义外,矢量的加、减和乘都比标量的加、减、乘和除更加复杂。一个矢量A可用一条用箭头指示方向的线段来表示,线段长度表示矢量A的模A,箭头指向表示矢量A的方向,如图1.1所示。一个模为1的矢量称为单位矢量。取aA表示与A同方向的单位矢量,则有A=aAA,其中
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0012_0001.jpg?sign=1738841482-udbYXprN3R3h0O3XWEp0S3CxKSE27Swk-0-a23288d87ba1d076667e137283abc112)
1.矢量加、减法
两个矢量A和B可按平行四边形法则相加,其对角线表示合成矢量C=A+B,如图1.2所示。矢量加法服从交换律和结合律
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0012_0002.jpg?sign=1738841482-ARr7vUWmT8VWYELSA9fzwnCBjP04xaff-0-711c17aee9e8c61a932186e27bc88c5a)
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0012_0003.jpg?sign=1738841482-kcIuRGZSACzFpJPvx1kvSRFCYnQraRws-0-80284e8af55fc60c9771bc03a01155f1)
B和-B可以看做大小相等方向相反的两个矢量,故借助于矢量加法也可以实现矢量减法,如图1.3所示,有
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0012_0004.jpg?sign=1738841482-2DNwOGnDanDjxTBJOHYhyGT40uSKJxY6-0-02a58d309a2a6877539d3a3bc0ffbd11)
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0012_0005.jpg?sign=1738841482-rNLTwEjvBjE526eeb0fUwgXHxclV9C4I-0-7f9be07dabe53dfbe0d3c14f0a966101)
图1.1 点P处的矢量
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0012_0006.jpg?sign=1738841482-SCxQukMMxoPUBrzCtUf9M11W0CrVf7gF-0-e4c920fcb61afe78213a9a7c2d068b7a)
图1.2 矢量加法
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0012_0007.jpg?sign=1738841482-9XVgUFK2QwqhwgXTgTm7HMAntrKNj48n-0-b7c9e835c5195bebfa0bfa569d3b5246)
图1.3 矢量减法
2.矢量乘法
一个标量η与一个矢量A的乘积ηA仍为一个矢量,其大小为|η|A,其方向由η的正负来确定:若η>0,则ηA与A平行同向;若η<0,则ηA与A平行反向。
两个矢量A和B的点积(或标积)A·B是一个标量,可看做两矢量相互投影之值,定义为
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0001.jpg?sign=1738841482-mkkLeZYINdYe89YrJgKHZrkifiKzxaUd-0-6a7b88aa931a7ce3bb98ddb4c93f9824)
式中,θ的取值范围为0≤θ≤π。如图1.4所示,当θ为锐角、直角和钝角时,点积标量为正、零和负值。矢量的点积满足交换律和分配律。
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0002.jpg?sign=1738841482-0XBnr77CxZFtKM8QtDv7yRp2zG3haLoA-0-32256d3be519551c1bc0bdea35f9ebb1)
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0003.jpg?sign=1738841482-7kfsTUrHCcNINAlw6AEkvqtWZr0g5K8Z-0-21f2724a27d47f9b555cc354502f4a60)
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0004.jpg?sign=1738841482-t7Y3jTSLx05moElZZJd5zK04xbj8HWGA-0-c6d63ef210e51c483b77185e28bf0b00)
图1.4 矢量点积
两个矢量A和B的叉积(或矢积)A×B是一个矢量,它垂直于A和B所在的平面,其指向按右旋法则来确定:当右手四指从矢量A旋转θ角至B时大拇指的指向,如图1.5所示,其定义为
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0005.jpg?sign=1738841482-ddXfu5e5JRhzTbQeTqHtgpOjbhcibhiZ-0-4951e121d755956356d8d5111003641b)
叉积不满足交换律,但满足分配律,有
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0006.jpg?sign=1738841482-n8lt8dD96KdPa7JLshFrMQh5qkoXQXax-0-36e2a2fa36f8beb1d99588f64e464c3a)
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0007.jpg?sign=1738841482-gwuzvtLzcF8MD4XB1puWLMjvkeTKBX2D-0-be0fa6ea8a9c08e9e35b20abec089d5b)
![](https://epubservercos.yuewen.com/1475A3/3590588604431801/epubprivate/OEBPS/Images/figure_0013_0008.jpg?sign=1738841482-NGBKfUXRv9FcpqJAwoSOv0JlcZnLcWB4-0-bf96bab0358229dbcc344145680c1e24)
图1.5 矢量叉积