Chapter 8 向量代数与空间解析几何
Section 1 向量及其线性运算
1.1 方向角和方向余弦
\(\overrightarrow{r} = \overrightarrow{OM} = (x,y,z)\)
称 \(\overrightarrow{r}\) 与坐标轴正向的夹角为方向角, 范围是 \([o,\pi]\)
\(cos\alpha , cos\beta , cos\gamma\) 为 \(\overrightarrow{r}\) 的方向余弦
\[cos\alpha = \frac{x}{\sqrt{x^2+y^2+z^2}}
\]
\[cos\beta = \frac{y}{\sqrt{x^2+y^2+z^2}}
\]
\[cos\gamma = \frac{z}{\sqrt{x^2+y^2+z^2}}
\]
\[\overrightarrow{e}_{\overrightarrow{r}} = (cos\alpha , cos\beta , cos\gamma)
\]
注意：
\[cos^2\alpha + cos^2\beta + cos^2\gamma = 1
\]
\[\overrightarrow{t} = (cos\alpha , cos\beta , cos\gamma) = (\frac{x}{|\overrightarrow{r}|} , \frac{y}{|\overrightarrow{r}|} , \frac{z}{|\overrightarrow{r}|})
\]
1.2 向量在轴上的投影
![[Pasted image 20260420150408.png]]
方法：
过 \(M\) 作与 \(u\) 垂直的平面 \(\pi\)
\(\pi\) 与 \(u\) 轴的交点为 \(M'\)
\(\overrightarrow{OM'}\) 是 \(\overrightarrow{OM}\) 在 \(u\) 轴的分向量
\(\overrightarrow{OM'} = \lambda \overrightarrow{e}\) , 则 \(\lambda\) 为 \(\overrightarrow{OM}\) 在 \(u\) 轴的投影
\[所以投影是一个数而并非向量
\]
性质：
\[①~Prj_{u}~\overrightarrow{OM} =|\overrightarrow{OM}|cos\varphi~~~~(\varphi \in [0,\pi])
\]
\[②~Prj_u~(\overrightarrow{a}+\overrightarrow{b}) = Prj_u~\overrightarrow{a} + Prj_u~\overrightarrow{b}
\]
\[③~Prj_u~\overrightarrow{\lambda a} = \lambda~Prj_u~\overrightarrow{a}
\]
\[④~
\left\{\begin{matrix}
Prj_\overrightarrow{b}~\overrightarrow{a} = |\overrightarrow{a}|cos\varphi \\
Prj_\overrightarrow{a}~\overrightarrow{b} = |\overrightarrow{b}|cos\varphi
\end{matrix}\right.\]
Section 2 数量积、向量积、混合积
数量积（点乘）
大小：
\[\overrightarrow{a}~\cdot~\overrightarrow{b} = |\overrightarrow{a}|\cdot|\overrightarrow{b}|\cdot cos\theta
\]
若 \(\overrightarrow{a} = (x_1,y_1,z_1)\) , \(\overrightarrow{b} = (x_2,y_2,z_2)\)
则
\[\overrightarrow{a}~\cdot~\overrightarrow{b} = x_1x_2 + y_1y_2 + z_1z_2
\]
其中夹角余弦：
\[cos\theta = \left\{\begin{matrix}
\frac{|\overrightarrow{a} \cdot \overrightarrow{b}|}{|\overrightarrow{a}| \cdot |\overrightarrow{b}|} \\
\frac{x_1x_2 + y_1y_2 + z_1z_2}{\sqrt{x_1^2+y_1^2+z_1^2} \cdot \sqrt{x_2^2+y_2^2+z_2^2}}
\end{matrix}\right.
\]
向量积（叉乘）
大小：
\[\overrightarrow{a}~\times~\overrightarrow{b} = |\overrightarrow{a}|\cdot|\overrightarrow{b}|\cdot sin\theta
\]
方向：右手准则 \(\Longrightarrow\) 从 \(a\) 往 \(b\) 绕，大拇指所指方向
矩阵计算：
\[\overrightarrow{a}~\times~\overrightarrow{b} =
\begin{vmatrix}\overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\x_1 & y_1 & z_1 \\x_2 & y_2 & z_2
\end{vmatrix}
\]
几何意义：三角形面积
\[S_{ABC} = \frac{1}{2}|\overrightarrow{a}\times\overrightarrow{b}|
\]
混合积
定义：
\[\left.\begin{matrix}
\overrightarrow{a} = (x_1,y_1,z_1) \\
\overrightarrow{b} = (x_2,y_2,z_2) \\
\overrightarrow{c} = (x_3,y_3,z_3)
\end{matrix}\right\}
\Longrightarrow
[\overrightarrow{a}~\overrightarrow{b}~\overrightarrow{c}] = (\overrightarrow{a} \times \overrightarrow{b})\cdot \overrightarrow{c} = |\overrightarrow{a} \times \overrightarrow{b}| \cdot |\overrightarrow{c}| \cdot cos\theta
\]
矩阵计算：
\[[\overrightarrow{a}~\overrightarrow{b}~\overrightarrow{c}] =
\begin{vmatrix}x_1 & y_1 & z_1 \\x_2 & y_2 & z_2 \\x_3 & y_3 & z_3
\end{vmatrix}
\]
几何意义：以三个向量为邻边的平行六面体的体积
\[V_{ABCDEFGH} = |[\overrightarrow{a}~\overrightarrow{b}~\overrightarrow{c}]| = |(\overrightarrow{a} \times \overrightarrow{b})\cdot \overrightarrow{c}| = |\overrightarrow{a} \times \overrightarrow{b}| \cdot |\overrightarrow{c}| \cdot |cos\theta|
\]
性质：
\[[[\overrightarrow{a}~\overrightarrow{b}~\overrightarrow{c}]] = [\overrightarrow{b}~\overrightarrow{c}~\overrightarrow{a}] = [\overrightarrow{c}~\overrightarrow{a}~\overrightarrow{b}]
\]
Section 3 平面方程
点法式方程
平面法向量 \(\overrightarrow{n} = (A,B,C)\)
平面上有个确定的点 \(M_0(x_0,y_0,z_0)\)
对于平面上任意一点 \(M(x,y,z)\)
有点法式方程
\[A(x-x_0)+B(y-y_0)+C(z-z_0)=0
\]
一般式方程
由点法式方程
\[A(x-x_0)+B(y-y_0)+C(z-z_0)=0
\]
可以得到一般式方程
\[Ax+By+Cz+D=0
\]
其中可以得到法向量 \(\overrightarrow{n} = (A,B,C)\)
截距式方程
平面过 \(x,y,z\) 轴上各一点 \(P(a,0,0),Q(0,b,0),R(0,0,c)\)
得到截距式方程
\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1
\]
距离/夹角
点到平面距离：
\[d=|Prj_\overrightarrow{n}~\overrightarrow{P_1P_0}| = \frac{|\overrightarrow{P_1P_0} \cdot \overrightarrow{n}|}{\overrightarrow{n}} = \frac{|A(x_0-x_1)+B(y_0-y_1)+C(z_0-z_1)|}{\sqrt{A^2+B^2+C^2}}
\]
两平面夹角：
\[cos\theta = \frac{|\overrightarrow{n_1} \cdot \overrightarrow{n_2}|}{|\overrightarrow{n_1}| \cdot |\overrightarrow{n_2}|} = \frac{|A_1A_2+B_1B_2+C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2} \cdot \sqrt{A_2^2+B_2^2+C_2^2}}
\]
Section 4 空间直线及方程
4.1 直线方程
一般式方程
即两平面交线
\[\left\{\begin{matrix}
A_1x+B_1y+C_1z+D_1=0 \\
A_2x+B_2y+C_2z+D_2=0
\end{matrix}\right.
\]
点向式方程
过 \(M_0(x_0,y_0,z_0)\) 且与 \(\overrightarrow{s} = (m,n,p) \ne \overrightarrow{0}\) 平行的直线方程
\[\frac{x-x_0}{m} = \frac{y-y_0}{n} = \frac{z-z_0}{p}
\]
参数式方程
由点向式方程 \(\frac{x-x_0}{m} = \frac{y-y_0}{n} = \frac{z-z_0}{p} = t\)
得到
\[\left\{\begin{matrix}
x=x_0+mt \\
y=y_0+nt \\
z=z_0+pt
\end{matrix}\right.
\]
两点式方程
\[\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}
\]
4.2 各类夹角
两向量夹角 \(\theta \in [0,\pi]\)
\[\left\{\begin{matrix}
\overrightarrow{a} = (m_1,n_1,p_1) \\
\overrightarrow{b} = (m_2,n_2,p_2)
\end{matrix}\right.
\]
得到夹角
\[cos\theta = \left\{\begin{matrix}
\frac{|\overrightarrow{a} \cdot \overrightarrow{b}|}{|\overrightarrow{a}| \cdot |\overrightarrow{b}|} \\
\frac{m_1m_2 + n_1n_2 + p_1p_2}{\sqrt{m_1^2+n_1^2+p_1^2} \cdot \sqrt{m_2^2+n_2^2+p_2^2}}
\end{matrix}\right.
\]
两直线夹角 \(\varphi \in [0,\frac{\pi}{2}]\)
\[\left\{\begin{matrix}
l_1 : \frac{x-x_1}{m_1} = \frac{y-y_1}{n_1} = \frac{z-z_1}{p_1} \\
l_2 : \frac{x-x_2}{m_2} = \frac{y-y_2}{n_2} = \frac{z-z_2}{p_2}
\end{matrix}\right.
\]
得到夹角
\[cos\varphi = \frac{m_1m_2 + n_1n_2 + p_1p_2}{\sqrt{m_1^2+n_1^2+p_1^2} \cdot \sqrt{m_2^2+n_2^2+p_2^2}}
\]
直线与平面的夹角 \(\varphi \in [0,\frac{\pi}{2}]\)
\[\left\{\begin{matrix}
l : \frac{x-x_0}{m} = \frac{y-y_0}{n} = \frac{z-z_0}{p} \\
\overrightarrow{s} = (m,n,p)
\end{matrix}\right.
\]
\[\left\{\begin{matrix}
\pi : Ax+By+Cz+D=0 \\
\overrightarrow{n} = (A,B,C)
\end{matrix}\right.
\]
得到夹角
\[sin\varphi = |cos\theta| = \frac{|Am+Bn+Cp|}{\sqrt{A^2+B^2+C^2} \cdot \sqrt{m^2+n^2+p^2}}
\]
两平面夹角 \(\varphi \in [0,\frac{\pi}{2}]\)
\[\left\{\begin{matrix}
\pi_1 : A_1x+B_1y+C_1z+D_1=0 \\
\pi_2 : A_2x+B_2y+C_2z+D_2=0
\end{matrix}\right.
~~~~\Longrightarrow
\left\{\begin{matrix}
\overrightarrow{n_1} = (A_1,B_1,C_1) \\
\overrightarrow{n_2} = (A_2,B_2,C_2)
\end{matrix}\right.
\]
得到夹角
\[cos\varphi = \frac{|A_1A_2+B_1B_2+C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2} \cdot \sqrt{A_2^2+B_2^2+C_2^2}}
\]
4.3各类距离
点到点
\[\left\{\begin{matrix}
P_1(x_1,y_1,z_1) \\
P_2(x_2,y_2,z_2)
\end{matrix}\right.
\]
得到距离
\[d=|\overrightarrow{P_2P_1}|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}
\]
点到线
\[\left\{\begin{matrix}
P_0(x_0,y_0,z_0) \\
\frac{x-x_1}{m} = \frac{y-y_1}{n} = \frac{z-z_1}{p} \Longrightarrow P_1(x_1,y_1,z_1)~,~\overrightarrow{s} = (m,n,p)
\end{matrix}\right.
\]
得到距离
\[d=\frac{|\overrightarrow{P_1P_0}\times\overrightarrow{s}|}{|\overrightarrow{s}|}=\frac{|\overrightarrow{P_1P_0}|\cdot|\overrightarrow{s}|\cdot sin\theta}{|\overrightarrow{s}|} = |\overrightarrow{P_1P_0}| \cdot sin\theta
\]
点到面
\[\left\{\begin{matrix}
P_0(x_0,y_0,z_0) \\
Ax+By+Cz+D=0 \Longrightarrow \overrightarrow{n} = (A,B,C)
\end{matrix}\right.
\]
得到距离
\[d=\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}
\]
线到线
\[\left\{\begin{matrix}
l_1 : \frac{x-x_1}{m_1} = \frac{y-y_1}{n_1} = \frac{z-z_1}{p_1} \\
l_2 : \frac{x-x_2}{m_2} = \frac{y-y_2}{n_2} = \frac{z-z_2}{p_2}
\end{matrix}\right.
\]
得到距离
\[d=\frac{|(\overrightarrow{s_1} \times \overrightarrow{s_2}) \cdot \overrightarrow{P_1P_2}|}{|\overrightarrow{s_1} \times \overrightarrow{s_2}|}
\]
面到面
\[\left\{\begin{matrix}
Ax+By+Cz+D_1=0 \\
Ax+By+Cz+D_2=0
\end{matrix}\right.
\]
得到距离
\[d=\frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}
\]