数学必修第二册：平面向量及其应用（一）
2023-08-09 16:31:56 8 举报AI智能生成
平面向量是在二维平面内既有方向又有大小的量，物理学中也称作矢量，与之相对的是只有大小、没有方向的数量（标量）。平面向量用a,b,c上面加一个小箭头表示，也可以用表示向量的有向线段的起点和终点字母表示。
作者其他创作
大纲/内容
平面向量的运算
向量加法
三角形法则
三角形法则就是<span class="equation-text" data-index="0" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="1" data-equation="\overrightarrow{BC}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overrightarrow{AC}"><span></span><span></span></span>这种计算法则叫做向量加法的三角形法则，简记为：首尾相连、连接首尾、指向终点
平行四边形法则
交换律
<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}+\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{b}+\overrightarrow{a}"><span></span><span></span></span>
结合律
<span class="equation-text" data-index="0" data-equation="(\overrightarrow{a}+\overrightarrow{b})" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="1" data-equation="\overrightarrow{c}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" contenteditable="false" data-index="3" data-equation="(\overrightarrow{b}+\overrightarrow{c})"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{b}"><span></span><span></span></span>共线同向
<span class="equation-text" data-index="0" data-equation="\overrightarrow{OA}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>,<span class="equation-text" data-index="2" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="3" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>,则<span class="equation-text" data-index="4" data-equation="\overrightarrow{a}+\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="5" data-equation="\overrightarrow{OB}"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{b}"><span></span><span></span></span>共线反向
<span class="equation-text" data-index="0" data-equation="\overrightarrow{OA}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>,<span class="equation-text" data-index="2" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="3" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>,则<span class="equation-text" data-index="4" data-equation="\overrightarrow{a}+\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="5" data-equation="\overrightarrow{OB}"><span></span><span></span></span>
等号成立，共线同向
|<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="1" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>|≤|<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>|+|<span class="equation-text" contenteditable="false" data-index="3" data-equation="\overrightarrow{b}"><span></span><span></span></span>|
向量减法
<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>和<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{b}"><span></span><span></span></span>互为相反向量
<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="-\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>,<span class="equation-text" data-index="2" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=-<span class="equation-text" data-index="3" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>,<span class="equation-text" data-index="4" data-equation="\overrightarrow{a}+\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="5" data-equation="\overrightarrow{0}"><span></span><span></span></span>
三角形法则
首首连，尾尾连，指向被减数向量
等号成立，共线反向
|<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="1" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>|≥|<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>|-|<span class="equation-text" contenteditable="false" data-index="3" data-equation="\overrightarrow{b}"><span></span><span></span></span>|
向量数乘
一般地， 我们规定实数λ与向址a的积是一个向量，这种运算叫做向量的数乘，记作λa
Iλal=Iλ||a|
当λ=0时，λa=0
当λ>0时，λa的方向与a的方向相同；当λ<0时，λa的方向与a的方向相反
(-1)a=-a
平面向量的概念
向量
概念
既有大小，又有方向的量
向量表示
用有向线段<span class="equation-text" data-index="0" data-equation="\overrightarrow{A B}" contenteditable="false"><span></span><span></span></span>或<span class="equation-text" data-index="1" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overrightarrow{b}"><span></span><span></span></span>来表示
数量
概念
只有大小，没有方向的量
数量表示
数轴上的点
有向线段
具有方向的线段
记作<span class="equation-text" data-index="0" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{AC}"><span></span><span></span></span>
有向线段的长度：|<span class="equation-text" data-index="0" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>| 、|<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{AC}"><span></span><span></span></span>|
三要素：起点、方向、长度
向量的大小
向量的长度或向量的模
记作：|<span class="equation-text" data-index="0" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>| 、|<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{a}"><span></span><span></span></span>|
零向量：长度为0的向量，方向任意，记作|<span class="equation-text" data-index="0" data-equation="\overrightarrow{0}" contenteditable="false"><span></span><span></span></span>|，|<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{0}"><span></span><span></span></span>|=0
单位向量：长度等于1单位长度的向量，<span class="equation-text" contenteditable="false" data-index="0" data-equation="|\overrightarrow{e}|"><span></span><span></span></span>=1
平行向量
<span style="font-size: inherit;">又称共线向量，方向相同或相反的非零向量，记作</span><span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(245, 245, 245); color: rgb(75, 75, 75); font-family: arial, 黑体; text-align: left; font-style: normal; font-weight: normal; display: inline !important; float: none;">//</span><span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{b}"><span></span><span></span></span>
零向量和任意向量平行
相等向量
长度相等，方向相同的向量，记作<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{a}=\overrightarrow{c}"><span></span><span></span></span>
相反向量
长度相等，方向相反的向量<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{a}=\overrightarrow{-d}"><span></span><span></span></span>
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