- 五年制高职数学(第二册)
- 赵春芳 王小燕 魏志丹
- 987字
- 2021-03-25 09:47:29
6.3 向量的数量积及其运算法则
本节重点知识:
1.向量的数量积.
2.向量数量积的坐标运算.
6.3.1 向量的数量积
在物理学中,一个物体在力的作用下,产生位移
,若
与
之间的夹角为θ,则
所作的功W是
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00029017.jpg?sign=1734442040-ONjhzf4pW5lRFhOUteDvmYmdWcvVjf5C-0-a83aa9a4c861360784fe17f3fe1e3025)
这里功W是一个数量,它由向量和
的模及其夹角余弦的乘积来确定.像这样由两个向量的模及其夹角余弦的乘积确定一个数量的情况,在其他一些问题中也会遇到,如物理学中的功率
等.
若将两个非零向量,
,设为
则把射线OA与射线OB所组成的不大于π的角称做
与
的夹角,记做
显然
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00029027.jpg?sign=1734442040-DSoh4VHp8RBjeYOLO8fl5YDywVgE9k29-0-4cc252a9919dedeb48631fbe4db0d754)
在数学中,我们将两个非零向量的模与它们的夹角θ的余弦的乘积定义为
与
的数量积(又称做内积),记做
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00029031.jpg?sign=1734442040-wWKjzTJkgfRPXCJgHymHdd7ivRyBqv61-0-45e1e086cf777cfccd4115b974e19f20)
其中θ表示
从而也可以表示成
注意 两个向量数量积的结果是一个实数,可能是正数,可能是负数,也可能是零.
想一想
如果 是两个非零向量,那么在什么条件下有以下结论:
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00030004.jpg?sign=1734442040-uwmzcaJxQn4qsEKeB77Zv0QFguSycPnR-0-c83b31cea1950a424d6f1955315831c6)
练一练
(1)如果 ,那么
_________;
(2)如果 ,那么
_________.
例1 根据下列条件分别求出
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00030010.jpg?sign=1734442040-rpryi8pmGQRwjP2FOOr9WyIypyasZAIk-0-66a2753bcaf2b925014889d6a515ea9e)
解 (1)因为
将已知条件代入,得
所以
又因为
所以
(2)因为
将已知条件代入,得
所以
又因为
所以
向量的数量积运算满足交换律和分配律,即
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00030021.jpg?sign=1734442040-CQh17ccq2PZ7UqM314qIVM3ucOmzEZ2C-0-95cd1e0bb6869fa7eacdeae04dbe218d)
但它不满足结合律,即
当实数与向量相乘时,满足结合律,即
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031001.jpg?sign=1734442040-kYC4Ywz5T7m0TRyGGnZjJZkohaTOYWMN-0-5eeef0888351bc51cbc68c195061770a)
例2 已知计算:
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031003.jpg?sign=1734442040-HfUs0sN0hMLyL4zyllfZSsdHJdQiU6Qg-0-a3cf59a6612ea3c4bffc4c9a00070977)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031004.jpg?sign=1734442040-ySsgaWDi3PQNghhpOlc1h2lUU69YfleJ-0-0f501ba37df5a12206c40086f1a9f0a9)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031005.jpg?sign=1734442040-MlEj4UjizOYAWVWlfYNx0ZhyaAratvhs-0-beede33697697da5ca9f6777ce2e77d6)
练习
1.已知分别是平面直角坐标系中x轴和y轴上的单位向量,分别计算:
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031007.jpg?sign=1734442040-0oTYCahZwxQkgObxfZ9otWOebXn6Rb4W-0-621cf8a5664dca75c13721503cc84180)
2.根据下列条件,求:
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031009.jpg?sign=1734442040-CtKxXacqeGr8t3LbyLQSmMsKvDldvtOj-0-c10bd5bb2dab016e522b9d29514424e7)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031010.jpg?sign=1734442040-o4jPK2DV7B4G7XYHAXquCPWyhjlTOBEo-0-1851b3fc8b9abd35872ab1c1c74056c6)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031011.jpg?sign=1734442040-EyEZ6JC0jEiLxuzdOm9HbIjvtGNkxOUS-0-6d10a0988bdbaaac7e7af8c290423539)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031012.jpg?sign=1734442040-6jw7zm3qwQKhOw06EvCopXLRGxOZDRD7-0-db2843b9b8c2a6b037d0a305a356c729)
3.已知求
4.已知计算:
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031016.jpg?sign=1734442040-MTOaDxWc8YPU3vTAF9BLmUsdSUwNLa5a-0-7332c7b39359facf213e70e50b32e82d)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031017.jpg?sign=1734442040-2NacH0ZupsrW9IT2TIypLzkEiUMnRyzj-0-359f83cae98351ef2a77709c80d34a45)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031018.jpg?sign=1734442040-vj7tR6B5dFIYhU9FMB9LF5pNhqc5rKwH-0-15326e12521489c63fdc829b9352783c)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031019.jpg?sign=1734442040-Mk6tD7REFUzYiCMhG3LKNHDxpmECSg0v-0-f8782866ad59624b89ac70ed749c74c2)
6.3.2 向量数量积的坐标运算
设向量的坐标为(x1,y1),即
向量
的坐标为(x2,y2)即
则
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031026.jpg?sign=1734442040-ZHViChjCVgesavO99bxDlk2M4eC0ZYyo-0-4c638c7082d02539a4e268c540dee097)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031027.jpg?sign=1734442040-S6YLRIq1mSk5f2gnENXR4RL6MZOinZdH-0-3725196dc70cd3ac8453540a542de852)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00031028.jpg?sign=1734442040-7281idmkQskipZ3gg090mdzqsQDM8uIr-0-1911b7bd497da14ad14f27ad02075158)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032001.jpg?sign=1734442040-tSGvsyXGjzMfNNHvr3j60SeB7EZTz3gB-0-a9ec7172221b2b2e56e83753ed14de5c)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032002.jpg?sign=1734442040-RG3WzSZHMewnTps3MDB2Uc2D3fGe56wg-0-a5b037b0cd0a805b724aae42b1c10147)
所以
就是说,在直角坐标系中,两个向量的数量积等于它们的横坐标之积与纵坐标之积的和.
例1 已知求
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032006.jpg?sign=1734442040-gjE6BTjacpIIOXxLagwbKP9hxrus7VUp-0-7f2639a39b478b7405adf422e8950411)
当两个向量垂直时,夹角为,此时有
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032008.jpg?sign=1734442040-Fu0fKgA9olN5gMjvhice8QF096FS1sGK-0-ad2f104241ac5295a32a41f9f1c84674)
反之,若非零向量的数量积为0,即
则必然有cosθ=0,即
故有
如果则有
例2 判断下列各题中的向量与
是否垂直:
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032017.jpg?sign=1734442040-dfN4NfGInNuOwRaeRL1pSDZglBxNjzo2-0-4030211fa0eb4958a80048cfc17c0458)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032018.jpg?sign=1734442040-8Xip0A01zG5ld6NWiFJt1N36YyLmmIWc-0-03d20178ec59152f855a8502c565d5f0)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032019.jpg?sign=1734442040-rV747aanywy1lva3N6RH47zBywzZ5jz6-0-cc638ca6e8e817d986cb1677aa175e2a)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00032020.jpg?sign=1734442040-qoOsEL2kuI7HGybg0LqayOUVDwyJh0tm-0-a7ee997895412e1f9ca0e439c3b9043d)
所以 与
不垂直.
如果那么
所以
就是说,利用向量坐标,我们可以计算出它的模.
练一练
算出下列各向量的模:
(1)若 ,则
(2)若 则
(3)若 则
如果点A坐标为(x1,y1),点B坐标为(x2,y2)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033001.jpg?sign=1734442040-YaNDxrmkyCihWpIJ8KfJDSzfRG7wGJY4-0-67d9426a78764b9365e70d7aae3dc701)
于是向量的模
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033003.jpg?sign=1734442040-K36PgGGrUsxxrHMLUA9Lu7brGOeINfwc-0-2711165938886e589d91525500d2565a)
由于的模就是点A和点B的距离,所以我们得到平面上两点间的距离公式
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033005.jpg?sign=1734442040-EHgYyHa1OZ2FSk4j9sr5iFvFLHJeYVtR-0-8b6da8a77cc53396d5966a8cf4070441)
例3 已知A(8,-1),B(2,7),求.
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033007.jpg?sign=1734442040-Oxj6nZQks1gnxKJ4xvs18Fcl3UaV6FFs-0-8e13ec678923a13479d082da82bca57c)
例4 已知点A(-3,-7),B(-1,-1),C(2,-2),求证:△ABC是直角三角形.
分析 可以通过判断某两边互相垂直,证得△ABC是直角三角形;也可以利用勾股定理的逆定理证得结论.
证法1:根据已知,得
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033008.jpg?sign=1734442040-YAP7UDAr1tuCzHhgkKjDgtxyVnj761Qx-0-ca318edd636a3368884de4f78a880a6b)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033009.jpg?sign=1734442040-nNlPgpCnv7V5kosNHp4VSZpShvsCGIxP-0-25d4ab51100d66e684a7299212a0f4ca)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033010.jpg?sign=1734442040-7CTbtSmnnrkWjESom3rJkP5twBv2D8Lo-0-a5220863969a76f8a0f8cc264cc918af)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033011.jpg?sign=1734442040-ZQrvUmyHZaYOtGY4sWkzUdWsOZndVnj3-0-4bfddb13ef1af95e456c49f697060447)
即∠ABC=90°.所以△ABC是直角三角形.
证法2:根据已知,得
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033012.jpg?sign=1734442040-iMRYLVzUn6YWijG9P9fjcKi2QfO7yVjI-0-cfcc16cbc80d0e57b0ef17291dda2dad)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033013.jpg?sign=1734442040-wAfLa3QizuKU3qKJHXeLLjeT8W4lbO8c-0-2c829b04d68f67297f5980884cd09557)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033014.jpg?sign=1734442040-ahPy71dycUCZE5e9z3Pq2nC68ocbRY5U-0-dbb5c6e58ac555dd448b342690ea4d70)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033015.jpg?sign=1734442040-AU7xZhbKBncqzGOUemGdXoj1QLr38Yy3-0-8d8c52d1ffe065cd9c0cbe4bc060f594)
即 CA2=AB2+BC2.所以△ABC是直角三角形.
练习
1.求的值,当:
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033018.jpg?sign=1734442040-5mdOJXlNVRPPHb9zjebjolJs72DTNjC2-0-71c79e538f7938e8d4d9e20a7a15a950)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033019.jpg?sign=1734442040-Lq9BuvEac72S5vzBsSv46b9lQDSlp1Ti-0-f613a3d85e92a6343211c0a6242909dc)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033020.jpg?sign=1734442040-iTuDmF8JW5uQfpNecQYXWglKHHXkUt0S-0-c0fbc65d53cc5a1f78044524a91194ed)
![](https://epubservercos.yuewen.com/689106/17180252105304906/epubprivate/OEBPS/Images/img00033021.jpg?sign=1734442040-5feJLPBaz14TlTIQvUk1kHLYmvq4pHVL-0-0b26942e0afadacd9937929f029d5b21)
2.已知M(6,4),N(1,-8),求
3.已知A(-4,7),B(5,-5),求