平面向量基本公式

<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{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

已知两个从同一点A出发的两个向量<span class="equation-text" data-index="0" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span>、<span class="equation-text" data-index="1" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>，以AC、AB为邻边作平行四边形ACDB，则以A为起点的对角线<span class="equation-text" data-index="2" data-equation="\overrightarrow{AD}" contenteditable="false"><span></span><span></span></span>就是向量<span class="equation-text" data-index="3" data-equation="\overrightarrow{AC}" contenteditable="false"><span></span><span></span></span>、<span class="equation-text" data-index="4" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>的和，这种计算法则叫做向量加法的平行四边形法则，简记为：共起点、对角连。

对于零向量和任意向量<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>，有：<span class="equation-text" data-index="1" data-equation="\overrightarrow{0}" 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" data-index="3" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="4" data-equation="\overrightarrow{0}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="5" data-equation="\overrightarrow{a}"><span></span><span></span></span>。

向量的加法满足所有的加法运算定律，如：交换律、结合律

<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" data-index="10" 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;">。</span><br>

<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{AC}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overrightarrow{CB}"><span></span><span></span></span>，这种计算法则叫做向量减法的三角形法则，简记为：共起点、连中点、指被减。

-(-<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{a}" 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" 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}" contenteditable="false"><span></span><span></span></span>)+<span class="equation-text" data-index="5" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="6" data-equation="\overrightarrow{0}" contenteditable="false"><span></span><span></span></span>；<span class="equation-text" data-index="7" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>-<span class="equation-text" data-index="8" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="9" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>+(-<span class="equation-text" contenteditable="false" data-index="10" data-equation="\overrightarrow{b}"><span></span><span></span></span>)。

<span style="font-size: inherit;">实数λ与向量</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(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;">的积是一个向量，这种运算叫做向量的数乘，记作λ</span><span class="equation-text" data-index="1" data-equation="\overrightarrow{a}" 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;">。当λ&gt;0时，λ</span><span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" 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;">的方向和<span class="equation-text" data-index="3" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的方向相同，当λ&lt;0时，λ<span class="equation-text" data-index="4" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的方向和<span class="equation-text" data-index="5" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的方向相反，当λ=0时，λ<span class="equation-text" data-index="6" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="7" data-equation="\overrightarrow{0}"><span></span><span></span></span>。</span><br>

设λ、μ是实数，那么满足如下运算性质

已知两个非零向量<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="1" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>, 它们的夹角为θ, 我们把数量l<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>I l<span class="equation-text" data-index="3" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>Icos⁡θ叫做向量<span class="equation-text" data-index="4" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" contenteditable="false" data-index="5" data-equation="\overrightarrow{b}"><span></span><span></span></span>的数量积

零向量与任意向量的数量积为0。

<span style="font-size: inherit;">数量积<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>的几何意义是：<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的长度|<span class="equation-text" data-index="3" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>|与<span class="equation-text" data-index="4" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="5" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的方向上的投影|<span class="equation-text" contenteditable="false" data-index="6" data-equation="\overrightarrow{b}"><span></span><span></span></span>|cos&nbsp;θ的乘积。</span><br>

数量积具有以下性质