平面向量运算
2023-08-10 09:39:03 6 举报AI智能生成
向量同数量一样，也可以进行运算。向量可以参与多种运算过程，包括线性运算（加法、减法和数乘）、数量积、向量积与混合积等。
作者其他创作
大纲/内容
向量的加法运算
概念：求两个向量和的运算，叫做向量的加法
方法
三角形法则
首尾相连，起点指终点
平行四边形法则
共起点，对角线
规定：对于零向量与任意向量<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{a}"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\overrightarrow{0}" 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{0}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="4" data-equation="\overrightarrow{a}"><span></span><span></span></span>。
｜|<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>｜
共线且反向取
|<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>|
共线且同向时取
向量的减法运算
规定：与向量<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>长度相等，方向相反的向量，叫做<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{a}"><span></span><span></span></span>的相反向量
记作-<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{a}"><span></span><span></span></span>
-(-<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" 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>
概念：求两个向量差的运算叫做向量的减法
减去一个向量相当于加上这个向量的相反向量
向量的数乘运算
定实数λ与向量<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{a}"><span></span><span></span></span>的积是一个向量，这种运算叫做向量的数乘
记作λ<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{a}"><span></span><span></span></span>
｜λ<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>｜=｜λ<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{a}"><span></span><span></span></span>｜
当λ>0时，λ<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的方向与<span class="equation-text" data-index="1" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的方向相同
当λ<0时，λ<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>的方向与<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{a}"><span></span><span></span></span>的方向相反
运算律
(λμ)<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>= λ(μ<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{a}"><span></span><span></span></span>)
(λ + μ)<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>= λ<span class="equation-text" data-index="1" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>+ μ<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overrightarrow{a}"><span></span><span></span></span>
λ(<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>) = λ<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>± λ<span class="equation-text" contenteditable="false" data-index="3" data-equation="\overrightarrow{b}"><span></span><span></span></span>
(－λ)<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>=－(λ<span class="equation-text" data-index="1" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>) = λ(－<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overrightarrow{a}"><span></span><span></span></span>)
向量<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>(a≠0)与<span class="equation-text" data-index="1" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>共线的充要条件是：存在唯一一个实数λ，使<span class="equation-text" data-index="2" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=λ<span class="equation-text" contenteditable="false" data-index="3" data-equation="\overrightarrow{a}"><span></span><span></span></span>
向量的数量积
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>, <span class="equation-text" data-index="1" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span> , O是平面上的任意一点，作<span class="equation-text" data-index="2" data-equation="\overrightarrow{OA}" 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{OB}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="5" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>，则∠AOB=θ，（0≤θ≤π）叫做向量<span class="equation-text" data-index="6" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" contenteditable="false" data-index="7" data-equation="\overrightarrow{b}"><span></span><span></span></span>的夹角<br>
θ=0时，<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>同向；θ=π时，<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overrightarrow{a}"><span></span><span></span></span>与<span class="equation-text" data-index="3" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>反向
θ=90°，我们说<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>垂直
记作<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>⊥<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow{b}"><span></span><span></span></span>
数量l<span class="equation-text" data-index="0" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>I l<span class="equation-text" data-index="1" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>Icosθ叫做向量<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="3" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>的数量积(或内积)，记作<span class="equation-text" contenteditable="false" data-index="4" data-equation="\overrightarrow{a}·\overrightarrow{b}"><span></span><span></span></span>
记作：<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{a}||\overrightarrow{b}|·cosθ"><span></span><span></span></span>
规定：零向量与任一向量的数量积为0
运算律
<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>
<span style="font-size: inherit;">(λ<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{a}"><span></span><span></span></span>)·<span class="equation-text" data-index="1" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>=λ(<span class="equation-text" data-index="2" data-equation="\overrightarrow{a}·\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>·(λ<span class="equation-text" data-index="4" data-equation="\overrightarrow{b}" contenteditable="false"><span></span><span></span></span>)</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" contenteditable="false" data-index="2" data-equation="\overrightarrow{c}"><span></span><span></span></span>=<span class="equation-text" data-index="3" data-equation="\overrightarrow{a}·\overrightarrow{c}" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="4" data-equation="\overrightarrow{b}·\overrightarrow{c}" contenteditable="false"><span></span><span></span></span>
0 条评论
下一页
为你推荐
查看更多
