数学必修第二册：平面向量及其应用（二）

设a, b是两个非零向量，<span class="equation-text" data-index="0" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>=a,<span class="equation-text" data-index="1" data-equation="\overrightarrow{CD}" contenteditable="false"><span></span><span></span></span>=b，我们考虑如下的变换，过<span class="equation-text" data-index="2" data-equation="\overrightarrow{AB}" contenteditable="false"><span></span><span></span></span>的起点A和终点B，分别做<span class="equation-text" data-index="3" data-equation="\overrightarrow{CD}" contenteditable="false"><span></span><span></span></span>所在直线的垂线，垂足分别为<span class="equation-text" data-index="4" data-equation="A_1" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="5" data-equation="B_1" contenteditable="false"><span></span><span></span></span>，得到我们称<span class="equation-text" contenteditable="false" data-index="6" data-equation="\overrightarrow{A_1&nbsp;B_1}"><span></span><span></span></span>上述变换为向量a是向量b的投影

<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow{A_1B_1}"><span></span><span></span></span>叫做向量a在向量b上的投影向量

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>