平面向量

<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>

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>

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>

在直角坐标系内，我们分别取与x轴、y轴方向相同的两个单位向量i、j作为基底。

余弦定理： 三角形中任何一边的平方、 等于其他两边平方的和减去这两边与它们夹角的余弦的积的两倍

a²=b²+c²-2bc<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos⁡A"><span></span><span></span></span>

b²=c²+a²-2ca<span class="equation-text" data-index="0" data-equation="cos⁡B" contenteditable="false"><span></span><span></span></span>

c²=a²+b²-2ab<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos⁡C"><span></span><span></span></span>

正弦定理：在一个三角形中，各边和它所对角的正弦的比相等

<span class="equation-text" data-index="0" data-equation="\frac{a}{sinA}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\frac{b}{sinB}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="2" data-equation="\frac{c}{sinC}"><span></span><span></span></span>

<span style="font-size: inherit;">已知向量</span><span class="equation-text" data-index="0" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">、</span><span class="equation-text" data-index="1" data-equation="\overrightarrow{BC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">，再作向量</span><span class="equation-text" data-index="2" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">，则向量</span><span class="equation-text" data-index="3" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">叫做</span><span class="equation-text" data-index="4" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">、</span><span class="equation-text" data-index="5" data-equation="\overrightarrow{BC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">的和，记作</span><span class="equation-text" data-index="6" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">+</span><span class="equation-text" data-index="7" data-equation="\overrightarrow{BC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">，即有：</span><span class="equation-text" data-index="8" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">+</span><span class="equation-text" data-index="9" data-equation="\overrightarrow{BC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">=</span><span class="equation-text" contenteditable="false" data-index="10" data-equation="\overrightarrow{AC}"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">。</span><br>

<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>

<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{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

等号成立，共线同向

<span style="font-size: inherit;">已知向量</span><span class="equation-text" data-index="0" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">、</span><span class="equation-text" data-index="1" data-equation="\overrightarrow{BC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">，再作向量</span><span class="equation-text" data-index="2" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">，则向量</span><span class="equation-text" data-index="3" data-equation="\overrightarrow{CB}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">叫做</span><span class="equation-text" data-index="4" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">、</span><span class="equation-text" data-index="5" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">的差，记作</span><span class="equation-text" data-index="6" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>-<span class="equation-text" data-index="7" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">，即有：</span><span class="equation-text" data-index="8" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>-<span class="equation-text" data-index="9" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">=</span><span class="equation-text" contenteditable="false" data-index="10" data-equation="\overrightarrow{CB}"><span></span><span></span></span><span style="font-size: inherit; background-color: rgb(255, 255, 255); color: rgb(13, 11, 34); font-family: 微软雅黑; font-weight: 400; text-align: left; font-style: normal; display: inline !important; float: none;">。</span><br>

|<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>|

Iλal=Iλ||a|

(-1)a=-a