浙江省专升本-空间解析几何与向量代数
2022-04-18 21:02:11 0 举报AI智能生成
浙江省专升本-空间解析几何与向量代数
作者其他创作
大纲/内容
向量代数的计算
向量计算
线性计算
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \pm \overrightarrow {b}=\{x_{1}\pm x_{2},y_{1}\pm y_{2},z_{1}\pm z_{2}\}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="k\overrightarrow {a}=\{kx_{1},ky_{1},kz_{1}\}"><span></span><span></span></span>
两点间距离
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|AB|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="向量的坐标和向量的坐标表示式"><span></span><span></span></span>
向量的坐标是一组有序数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x_1,y_1,z_1),(x_2,y_2,z_2)"><span></span><span></span></span>,<span class="equation-text" contenteditable="false" data-index="1" data-equation="x_1,y_1,z_1..."><span></span><span></span></span>为有序数
向量的坐标表示式是一个向量
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {M_1M_2} = (x_2-x_1)i+(y_2-y_1)j+(z_1-z_2)k"><span></span><span></span></span>
向径：r=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {OM}"><span></span><span></span></span>称为点M关于原点O的向径
<span class="equation-text" contenteditable="false" data-index="0" data-equation="M\Leftrightarrow r=\overrightarrow {OM}=xi+yj+zk \Leftrightarrow (x,y,z)"><span></span><span></span></span>
数量积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \cdot \overrightarrow {b} = |a||b|cos(\theta)= x_{1} x_{2}+ y_{1} y_{2}+z_{1}z_{2}=|\overrightarrow {b}| \Pr _{\overrightarrow {b}} \overrightarrow {a}=|\overrightarrow {a}| \Pr _{\overrightarrow {a}} \overrightarrow {b}"><span></span><span></span></span>
运算性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \cdot \overrightarrow{b} =\overrightarrow {b} \cdot \overrightarrow{a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \cdot (\overrightarrow {b} + \overrightarrow {c}) = \overrightarrow {a} \cdot \overrightarrow {b} +\overrightarrow {a} \cdot \overrightarrow {c}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lambda(\overrightarrow {a} \cdot \overrightarrow{b}) = (\lambda\overrightarrow {a}) \cdot \overrightarrow{b}=\overrightarrow {a} \cdot (\lambda\overrightarrow{b})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \cdot \overrightarrow {a} = |\overrightarrow {a}|^2,|\overrightarrow {a}|=\sqrt{\overrightarrow {a}\cdot \overrightarrow {a}}"><span></span><span></span></span>
向量积
含义
满足右手法则，<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {c}"><span></span><span></span></span>垂直于<span class="equation-text" contenteditable="false" data-index="1" data-equation="\overrightarrow {a}"><span></span><span></span></span>与<span class="equation-text" contenteditable="false" data-index="2" data-equation="\overrightarrow {b} 所确定的平面"><span></span><span></span></span>
运算性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \times\overrightarrow{b} =-\overrightarrow {b} \times\overrightarrow{a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \times(\overrightarrow {b} + \overrightarrow {c}) = \overrightarrow {a} \times\overrightarrow {b} +\overrightarrow {a} \times \overrightarrow {c}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lambda(\overrightarrow {a} \times\overrightarrow{b}) = (\lambda\overrightarrow {a}) \times\overrightarrow{b}=\overrightarrow {a} \times(\lambda\overrightarrow{b})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {c} = \overrightarrow {a} \times \overrightarrow {b}= \begin{vmatrix}\overrightarrow {i} & \overrightarrow {j} & \overrightarrow {k}\\x_{1} & y_{1} & z_{1}\\ x_{2} & y_{2} & z_{2}\end{vmatrix}=\begin{vmatrix}y_1 & z_1 \\y_2& z_2\end{vmatrix}\overrightarrow i-\begin{vmatrix}x_1 & z_1 \\x_2& z_2\end{vmatrix}\overrightarrow j+\begin{vmatrix}x_1 & y_1 \\x_2& y_2\end{vmatrix}\overrightarrow k"><span></span><span></span></span>
混合积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a,b,c)=(a\times b)\cdot c=\begin{vmatrix}\ x_{1} & y_{1} & z_{1}\\ x_{2} & y_{2} & z_{2}\\x_{3} & y_{3} & z_{3}\end{vmatrix}"><span></span><span></span></span>
运算性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a,b,c)=(b,c,a)=(c,a,b)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \cdot \overrightarrow {a} =|\overrightarrow {a}|^2,(\overrightarrow {a}\times\overrightarrow {b})\cdot \overrightarrow {a}=0,\overrightarrow {a} \times\overrightarrow {a} =0,"><span></span><span></span></span>
注意:<span class="equation-text" contenteditable="false" data-index="0" data-equation="|\cdot|不是绝对值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a,b,c)=-(a,c,b)=-(c,b,a)=-(b,a,c)"><span></span><span></span></span>
投影
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a}在\overrightarrow {b}"><span></span><span></span></span>向量上的投影为：<span class="equation-text" contenteditable="false" data-index="1" data-equation="\Pr _{\overrightarrow {b}} \overrightarrow {a} = \frac{\overrightarrow {a} \cdot \overrightarrow {b}}{|\overrightarrow {b}|}=|\overrightarrow {a}|cos\theta"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{\overrightarrow {a} \cdot \overrightarrow {b}}{|\overrightarrow {b}|} \Leftrightarrow \overrightarrow {a} \cdot \frac{\overrightarrow {b}}{|\overrightarrow {b}|}可以理解为\overrightarrow {a} 与单位向量\frac{\overrightarrow {b}}{|\overrightarrow {b}|}的数量积"><span></span><span></span></span>
向量平行四边形
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S=|\overrightarrow {a}||\overrightarrow {b}|sin\theta = |\overrightarrow {a} \times \overrightarrow {b}|"><span></span><span></span></span>
向量三角形
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S=\frac{1}{2} |\overrightarrow {a} \times \overrightarrow {b}|=\frac{1}{2} |\overrightarrow {a}||\overrightarrow {b}|sin\theta"><span></span><span></span></span>
方向角
含义
向量<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {OM}与x轴，y轴，z轴的夹角分别是\alpha,\beta,\gamma"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\alpha = \frac{x}{\sqrt{x^2} +y^2+z^2},cos\beta=\frac{y}{\sqrt{x^2} +y^2+z^2},cos\gamma=\frac{z}{\sqrt{x^2} +y^2+z^2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(cos\alpha)^2+(cos\beta)^2+(cos\gamma)^2=1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha,\beta,\gamma\in[0,180^{\circ}]"><span></span><span></span></span>
单位向量
单位向量<span class="equation-text" contenteditable="false" data-index="0" data-equation="e _{\rightarrow }=(x,y,z),则有x^2+y^2+z^2=1"><span></span><span></span></span>
单位向量的模长等于1
模为0的向量为零向量，方向是任意的
与向量<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {b}"><span></span><span></span></span>同方向的单位向量
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{\overrightarrow {b}}{|\overrightarrow {b}|}"><span></span><span></span></span>
与向量<span class="equation-text" data-index="0" data-equation="\overrightarrow {b}" contenteditable="false"><span></span><span></span></span>平行的单位向量
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\pm \frac{\overrightarrow {b}}{|\overrightarrow {b}|}"><span></span><span></span></span>
过<span class="equation-text" contenteditable="false" data-index="0" data-equation="xOy...平面与向量\overrightarrow {b}垂直的单位向量"><span></span><span></span></span>
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x,y,0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="利用单位向量性质x^2+y^2=1"><span></span><span></span></span>
利用向量垂直性质<span class="equation-text" contenteditable="false" data-index="0" data-equation="ax+by=0"><span></span><span></span></span>
点到坐标系的距离
点<span class="equation-text" contenteditable="false" data-index="0" data-equation="M(x_0,y_0,z_0)"><span></span><span></span></span>（缺谁写谁）<br>
到原点<span class="equation-text" contenteditable="false" data-index="0" data-equation="O"><span></span><span></span></span>距离
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=\sqrt{x_0^2+y_0^2+z_0^2}"><span></span><span></span></span>
到<span class="equation-text" contenteditable="false" data-index="0" data-equation="OY"><span></span><span></span></span>轴的距离
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=\sqrt{x_0^2+z_0^2}"><span></span><span></span></span>
到<span class="equation-text" contenteditable="false" data-index="0" data-equation="YOZ"><span></span><span></span></span>平面的距离
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=\sqrt{x_0^2}=|x_0|"><span></span><span></span></span>
绝对值三角不等式
同向平行向量有
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|a+b|=|a|+|b|"><span></span><span></span></span>
反向平行向量有
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|a+b|=|a|-|b|"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|a|-|b|\leq |a\pm b| \leq |a|+|b|"><span></span><span></span></span>
向量平行与垂直
向量垂直
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} \perp \overrightarrow {b} \Leftrightarrow \overrightarrow {a} \cdot \overrightarrow {b} =0 \Leftrightarrow x_{1} x_{2}+ y_{1} y_{2}+z_{1}z_{2}=0"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="\Leftrightarrow \overrightarrow {a} = \lambda \overrightarrow {b} "><span></span><span></span></span>
向量平行
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a} //\overrightarrow {b} \Leftrightarrow \overrightarrow {a} \times \overrightarrow {b}=0 \Leftrightarrow \frac{x_{1}}{x_{2}}=\frac{y_{1}}{y_{2}}=\frac{z_{1}}{z_{2}}=\lambda"><span></span><span></span></span>
重要结论
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\forall A(x_1,y_1,z_1),B(x_2,y_2,z_2),if \overrightarrow {AM} = \lambda \overrightarrow {MB}，s.t. \overrightarrow {OM} =(\frac{x_1+\lambda x_2}{1+\lambda},\frac{y_1+\lambda y_2}{1+\lambda},\frac{z_1+\lambda z_2}{1+\lambda})"><span></span><span></span></span>
共线、共面
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a},\overrightarrow {b}共线 \Leftrightarrow \exists 不全为零的数\lambda,\mu,s.t. \lambda\overrightarrow {a}+\mu\overrightarrow {b}=0"><span></span><span></span></span>
证明不重合的三点<span class="equation-text" contenteditable="false" data-index="0" data-equation="A,B,C"><span></span><span></span></span>共线，只需要证明<span class="equation-text" contenteditable="false" data-index="1" data-equation="| \overrightarrow {AB} \times \overrightarrow {BC} | =0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overrightarrow {a},\overrightarrow {b},\overrightarrow {c} 共面\Leftrightarrow \exists 不全为零的数\lambda,\mu,\gamma, s.t. \lambda\overrightarrow {a}+\mu\overrightarrow {b}+\gamma\overrightarrow {c}=0或(a,b,c)=0"><span></span><span></span></span>
向量夹角
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\theta = \frac{a \cdot b}{|a|\cdot |b|}=\frac{ x_{1} x_{2}+ y_{1} y_{2}+z_{1}z_{2}}{\sqrt{x_{1}^2+y_{1}^2+z_{1}^2}+\sqrt{x_{2}^2+y_{2}^2+z_{2}^2}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{a\cdot b}{|a||b|} \Leftrightarrow \frac{a}{|a|} \cdot \frac{b}{|b|}"><span></span><span></span></span>可以理解为两个单位向量的数量积
夹角范围
直线和直线
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[0,\frac{\pi}{2}]"><span></span><span></span></span>
直线和平面
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[0,\frac{\pi}{2}]"><span></span><span></span></span>
平面和平面
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[0,\frac{\pi}{2}]"><span></span><span></span></span>
三种积几何含义
数量积
两个向量夹角的角度
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\theta = \frac{a \cdot b}{|a|\cdot |b|}=\frac{ x_{1} x_{2}+ y_{1} y_{2}+z_{1}z_{2}}{\sqrt{x_{1}^2+y_{1}^2+z_{1}^2}+\sqrt{x_{2}^2+y_{2}^2+z_{2}^2}}"><span></span><span></span></span>
向量积
平行四边形的面积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S=|\overrightarrow {AB} \times \overrightarrow {BC}|"><span></span><span></span></span>
混合积
三个向量定义的平行六面体的体积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="V=|(\overrightarrow {a} \times \overrightarrow {b}) \cdot \overrightarrow {c}|"><span></span><span></span></span>
空间坐标系挂限
挂限图
罗马数字
<span class="equation-text" contenteditable="false" data-index="0" data-equation="I 、II、 III、 IV 、V、 VI、 VII、 VIII "><span></span><span></span></span>
平面
平面方程的求法
平面一般式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="Ax+By+Cz+D=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="D=0，平面通过坐标原点"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=0,\begin{cases}D=0,平面通过x轴 \\D\neq 0,平面平行于x轴\end{cases}"><span></span><span></span></span>
类似可讨论<span class="equation-text" contenteditable="false" data-index="0" data-equation="B=0,C=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=B=0,平面平行于xOy坐标面"><span></span><span></span></span>
类似可讨论<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=C=0,B=C=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=B=D=0,表示xOy坐标面"><span></span><span></span></span>
类似可讨论<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=C=D=0,B=C=D=0"><span></span><span></span></span>
平面点法式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0"><span></span><span></span></span>
平面截距式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{x}{a} +\frac{y}{b}+\frac{z}{c}=1"><span></span><span></span></span>其中a,b,c分别是x,y,z上的截距
平面三点式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{vmatrix}\ x-x_{1} & y-y_{1} & z-z_{1}\\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1}\end{vmatrix}=0"><span></span><span></span></span>
点到平面的距离
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=\frac{|Ax_{0}+By_{0}+Cz_{0}+D|}{\sqrt{A^2+B^2+C^2}}"><span></span><span></span></span>
平面到平面的距离(平行)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=\frac{|D_{1}-D_{2}|}{\sqrt{A^2+B^2+C^2}}"><span></span><span></span></span>
平面与平面垂直
互相垂直<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Leftrightarrow n_{1}\perp n_{2} \Leftrightarrow A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}=0"><span></span><span></span></span>
平面和平面相交夹角为<span class="equation-text" contenteditable="false" data-index="0" data-equation="\theta"><span></span><span></span></span>(通常锐角)
设平面<span class="equation-text" contenteditable="false" data-index="0" data-equation="\pi_1,\pi_2的法线向量依次为n_1=(A_1,B_1,C_1)和n_2=(A_2,B_2,C_2)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\pi_1,\pi_2的夹角\theta应该是(\widehat{n_1,n_2})和(-\widehat{n_1,n_2})=\pi-(\widehat{n_1,n_2})两者中的锐角，因此cos\theta=|cos(\widehat{n_1,n_2})|=\frac{ |A_{1} A_{2}+ B_{1} B_{2}+C_{1}C_{2}|}{\sqrt{A_{1}^2+B_{1}^2+C_{1}^2}+\sqrt{A_{2}^2+B_{2}^2+C_{2}^2}}"><span></span><span></span></span>
平面和平面的位置关系(充分必要条件)
垂直
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}=0"><span></span><span></span></span>
平行
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}"><span></span><span></span></span>
重合
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}=\frac{D_1}{D_2}"><span></span><span></span></span>
直线
直线方程的求法
直线一般式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}A_{1}x+B_{1}y+C_{1}z+D_{1}=0 \\A_{2}x+B_{2}y+C_{2}z+D_{2}=0 \end{cases}"><span></span><span></span></span>
直线(对称式)点法向式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{x-x_{0}}{l}=\frac{y-y_{0}}{m}=\frac{z-z_{0}}{n}"><span></span><span></span></span>
直线两点式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{x-x_{0}}{x_{1}-x_{0}}=\frac{y-y_{0}}{y_{1}-y_{0}}=\frac{z-z_{0}}{z_{1}-z_{0}}"><span></span><span></span></span>
直线参数式方程
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}x=x_{0}+lt\\y=y_{0} +mt \\z=z_{0}+nt \end{cases}"><span></span><span></span></span>
点到直线的距离
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=\frac{|\overrightarrow {P_{0}M_{1}} \times \overrightarrow s|}{|\overrightarrow {s}|}"><span></span><span></span></span>
异面直线间的距离
<span class="equation-text" contenteditable="false" data-index="0" data-equation="d=\frac{|\overrightarrow {P_{0}P_{1}} \cdot (\overrightarrow s_{1}\times \overrightarrow s_{2})|}{|\overrightarrow {s_{1}}\times \overrightarrow {s_{2}} |}"><span></span><span></span></span>
相交直线或重合的直线间的距离为0
直线和直线垂直
互相垂直 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\Leftrightarrow \overrightarrow {s_{1} } \perp \overrightarrow {s_{2} } \Leftrightarrow l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}=0"><span></span><span></span></span>
直线和直线间相交角<span class="equation-text" contenteditable="false" data-index="0" data-equation="\theta"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos\theta = \frac{|\overrightarrow {s_{1}} \cdot \overrightarrow {s_{2}}|}{|\overrightarrow {s_{1}} ||\overrightarrow {s_{2}}|}=\frac{|l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}|}{\sqrt{l_{1}^2+m_{1}^2+n_{1}^2} \cdot \sqrt{l_{2}^2+m_{2}^2+n_{2}^2}}"><span></span><span></span></span>
直线和平面相交角<span class="equation-text" contenteditable="false" data-index="0" data-equation="\gamma"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\gamma = \frac{|\overrightarrow {s} \cdot \overrightarrow {n}|}{|\overrightarrow {s} ||\overrightarrow {n}|}=\frac{|Al+Bm+Cn|}{\sqrt{A^2+B^2+C^2} \cdot \sqrt{l^2+m^2+n^2}}"><span></span><span></span></span>
直线和平面的关系题型
<span class="equation-text" contenteditable="false" data-index="0" data-equation="l\perp \pi \Leftrightarrow \overrightarrow {s} // \overrightarrow {n} = \begin{cases} l//\pi \Rightarrow \overrightarrow {s} \perp \overrightarrow {n}\\ \overrightarrow {s} \perp \overrightarrow {n} \Rightarrow l//\pi或l\subset \pi\end{cases}"><span></span><span></span></span>
直线和直线的相交角<span class="equation-text" contenteditable="false" data-index="0" data-equation="\theta"><span></span><span></span></span>（通常指锐角）
设平面<span class="equation-text" contenteditable="false" data-index="0" data-equation="L_1,L_2的法线向量依次为s_1=(m_1,n_1,p_1)和s_2=(m_2,n_2,p_2)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="L_1,L_2的夹角\theta应该是(\widehat{s_1,s_2})和(-\widehat{s_1,s_2})=\pi-(\widehat{s_1,s_2})两者中的锐角，因此cos\theta=|cos(\widehat{s_1,s_2})|=\frac{ |m_{1} m_{2}+ n_{1} n_{2}+p_{1}p_{2}|}{\sqrt{m_{1}^2+n_{1}^2+p_{1}^2}+\sqrt{m_{2}^2+n_{2}^2+p_{2}^2}}"><span></span><span></span></span>
直线和直线的位置关系(充分必要条件)
垂直
<span class="equation-text" contenteditable="false" data-index="0" data-equation="m_{1} m_{2}+ n_{1} n_{2}+p_{1}p_{2}=0"><span></span><span></span></span>
平行或重合
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{m_1}{m_2}=\frac{n_1}{n_2}=\frac{p_1}{p_2}"><span></span><span></span></span>
直线和平面的夹角<span class="equation-text" contenteditable="false" data-index="0" data-equation="\theta"><span></span><span></span></span>
设直线方向向量<span class="equation-text" contenteditable="false" data-index="0" data-equation="s=(m,n,p)，平面法线向量n=(A,B,C)直线与平面夹角\alpha"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha=|\frac{\pi}{2}-(\widehat{s,n})|，因此sin\alpha=sin|\frac{\pi}{2}-(\widehat{s,n})|=|cos(\widehat{s,n})|=\frac{ |Am+ Bn+Cp|}{\sqrt{A^2+B^2+C^2}+\sqrt{m^2+n^2+p^2}}"><span></span><span></span></span>
直线和平面的关系
垂直
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{A}{m}=\frac{B}{n}=\frac{C}{p}"><span></span><span></span></span>
平行或在平面上
<span class="equation-text" contenteditable="false" data-index="0" data-equation="Am+Bn+Cp=0"><span></span><span></span></span>
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