高数笔记 P06：向量代数与空间解析几何

1 向量

向量的模、方向角、投影

• \(|\boldsymbol r| = \sqrt{x^2 + y^2 + z^2}​\)
• 两点距\(|AB| = \sqrt{(x_1 - x_2)^2 + (y_2 - y_2)^2 + (z_1 + z_2)^2}\)
• 方向角：非零向量与三个座标轴的夹角\(\alpha, \beta, \gamma\)
• 方向余弦：\(\begin{cases} \cos \alpha = \cfrac x{|\boldsymbol r|} \\ \cos \beta = \cfrac y{|\boldsymbol r|} \\ \cos \gamma = \cfrac z{|\boldsymbol r|} \end{cases}\)
• 向量方向上的单位向量\(\boldsymbol e = (\cos \alpha , \cos \beta , \cos \gamma )\)，由此可得\(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\)
• \(u\)轴上的投影\(\mathrm{Prj}_u \boldsymbol r \ or \ (\boldsymbol r)_u\)
  • \(\mathrm{Prj}_u \boldsymbol a = |\boldsymbol a| \cos \varphi\)，\(\varphi\)为\(\boldsymbol a\)在\(u\)轴上的夹角。
  • \(\mathrm{Prj}_u (\boldsymbol a + \boldsymbol b) = \mathrm{Prj}_u \boldsymbol a + \mathrm{Prj}_u \boldsymbol b\)
  • \(\mathrm{Prj}_u \lambda \boldsymbol a = \lambda \mathrm{Prj}_u \boldsymbol a\)

向量代数运算

• 加减法：平行四边形法则，符合交换律、结合律。
• 数乘：符合结合律、分配律 ​

数量积、向量积、混合积

• 数量积 \(\boldsymbol a \cdot \boldsymbol b = |\boldsymbol a| |\boldsymbol b| \cos (\widehat{\boldsymbol a ,\boldsymbol b}) = |\boldsymbol a| \mathrm{Prj}_{\boldsymbol a} \boldsymbol b = |\boldsymbol b| \mathrm{Prj}_{\boldsymbol b} \boldsymbol a = a_x b_x + a_y b_y + a_z b_z\)
  • \(\boldsymbol a \cdot \boldsymbol a = |\boldsymbol a|^2\)
  • 交换律 \(\boldsymbol a \cdot \boldsymbol b = \boldsymbol b \cdot \boldsymbol a\)
  • 分配律 \((\boldsymbol a + \boldsymbol b) \cdot \boldsymbol c = \boldsymbol a \cdot \boldsymbol c + \boldsymbol b \cdot \boldsymbol c\)
  • \((\lambda \boldsymbol a) \cdot \boldsymbol b = \lambda(\boldsymbol a \cdot \boldsymbol b)\)
  • 两向量夹角\(\theta\)，由此可得\(\cos \theta = \cfrac {\boldsymbol a \cdot \boldsymbol b}{|\boldsymbol a| |\boldsymbol b|}\)
• 向量积 \(\boldsymbol a \times \boldsymbol b = \left| \begin{matrix} \boldsymbol i & \boldsymbol j & \boldsymbol k \\ a_x & a_y & a_z \\ b_x & b_y & b_z \\ \end{matrix} \right|\)，几何意义：同时垂直于\(\boldsymbol a, \boldsymbol b\)的向量，满足右手规则。
  • \(|\boldsymbol a \times \boldsymbol b| = |\boldsymbol a| |\boldsymbol b| \sin (\widehat{\boldsymbol a,\boldsymbol b})\) （这个也是\(\boldsymbol a, \boldsymbol b\)为邻边组成的平行四边形的面积。
  • \(\boldsymbol a \times \boldsymbol a = \boldsymbol 0\)
  • \(\boldsymbol b \times \boldsymbol a = - \boldsymbol a \times \boldsymbol b\)
  • 分配律 \((\boldsymbol a + \boldsymbol b) \times \boldsymbol c = \boldsymbol a \times \boldsymbol c + \boldsymbol a \times \boldsymbol c\)
  • \((\lambda \boldsymbol a) \times \boldsymbol b = \boldsymbol a \times (\lambda \boldsymbol b) = \lambda (\boldsymbol a \times \boldsymbol b)\)
• 混合积\([\boldsymbol a \boldsymbol b \boldsymbol c] = (\boldsymbol a \times \boldsymbol b) \cdot \boldsymbol c = \left| \begin{matrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \\ \end{matrix} \right|\)
  • 轮换对称性\([\boldsymbol a \boldsymbol b \boldsymbol c] = [\boldsymbol b \boldsymbol c \boldsymbol a] = [\boldsymbol c \boldsymbol a \boldsymbol b]\)
  • 两向量互换，混合积变号：\([\boldsymbol a \boldsymbol b \boldsymbol c] = -[\boldsymbol a \boldsymbol c \boldsymbol b] = -[\boldsymbol c \boldsymbol b \boldsymbol a] = -[\boldsymbol b \boldsymbol a \boldsymbol c]\)
  • 以\(\boldsymbol a, \boldsymbol b, \boldsymbol c\)为棱的平行六面体的体积：\(V = |[\boldsymbol a \boldsymbol b \boldsymbol c]|\)

• \(\boldsymbol a \parallel \boldsymbol b \iff\)存在唯一实数\(\lambda\)使\(\boldsymbol a = \lambda \boldsymbol b \iff \boldsymbol a \times \boldsymbol b = \boldsymbol 0\)
• \(\boldsymbol a \perp \boldsymbol b \iff \boldsymbol a \cdot \boldsymbol b = 0\)
• \(\boldsymbol a, \boldsymbol b, \boldsymbol c\)共面\(\iff [\boldsymbol a \boldsymbol b \boldsymbol c] = 0\)

2 平面与直线

曲面方程\(F(x, y, z) = 0\)

平面方程

• 一般式方程：\(Ax + By + Cz +D = 0,\quad \boldsymbol n = {A, B, C}\) 为平面的法向量，\(A, B, C\) 不全为零。
• 点法式方程：\(A(x - x_0) + B(y - y_0) + C(z - z_0) = 0\)，其中\((x_0, y_0, z_0)\)为平面上任一点。
• 截距式方程：\(\cfrac xa + \cfrac yb + \cfrac zc = 1\)，\(a, b, c\) 为平面在三个坐标轴的截距，全不为零。

直线方程

• 一般方程：\(\begin{cases} A_1 x + B_1 y + C_1 z +D_1 = 0 \\ A_2 x + B_2 y + C_2 z + D_2 = 0 \end{cases}\)
• 对称式方程：\(\cfrac {x - x_0}m = \cfrac {y - y_0}n = \cfrac {z - z_0}p\)，其中\((x_0, y_0, z_0)\)为直线上一点，\(\boldsymbol s = (m, n, p)\)为直线的方向向量。
• 参数式方程：\(\begin{cases} x = x_0 + mt \\ y = y_0 + nt \\ z = z_0 + pt \\ \end{cases}\)

平面、直线之间的关系

• 两平面$ A_1 x + B_1 y + C_1z + D_1 = 0, A_2 x + B_2 y + C_2 z +D_2 = 0$ 夹角\(\theta\)为两平面所成角度的直角或者锐角，满足：\(\cos \theta = |\cos (\widehat{\boldsymbol n_1, \boldsymbol n_2})| = \cfrac {|A_1 A_2 + B_1 B_2 + C_1 C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2} \sqrt{A_2^2 + B_2^2 + C_2^2}}\)
  • 两平面平行或者重合\(\theta = 0 \iff \cfrac {A_1}{A_2} = \cfrac {B_1}{B_2} = \cfrac {C_1}{C_2}\)
  • 两平面垂直\(\theta = \cfrac {\pi}2 \iff A_1 A_2 + B_1 B_2 + C_1 C_2 = 0\)
• 两直线\(\cfrac {x - x_1}{m_1} = \cfrac {y - y_1}{n_1} = \cfrac {z - z_1}{p_1}, \ \cfrac {x - x_2}{m_2} = \cfrac {y - y_2}{n_2} = \cfrac {z - z_2}{p_2}\) 的夹角\(\theta\)为两直线所成角度的直角或者锐角，满足：\(\cos \theta= |\cos (\widehat {\boldsymbol s_1, \boldsymbol s_2})| = \cfrac {|m_1 m_2 + n_1 n_2 + p_1 p_2|}{\sqrt{m_1^2 + n_1^2 + p_1^2} \sqrt{m_2^2 + n_2^2 + p_2^2}}\)
  • 两直线平行或者重合\(\theta= 0 \iff \cfrac {m_1}{m_2} = \cfrac {n_1}{n_2} = \cfrac {p_1}{p_2}\)
  • 两平面垂直\(\theta = \cfrac {\pi}2 \iff m_1 m_2 + n_1 n_2 + p_1 p_2 = 0\)
• 平面与直线的夹角\(\theta\)满足\(\sin \theta = |\cos (\widehat{\boldsymbol s, \boldsymbol n})| = \cfrac {|Am + Bn + Cp|}{\sqrt{A^2 + B^2 + C^2}{\sqrt{m^2 + n^2 + p^2}}}\)
  • 平面与直线平行\(\theta = 0 \iff \cfrac Am = \cfrac Bn = \cfrac Cp\)
  • 平面与直线垂直\(\theta = \cfrac {\pi}2 \iff Am + Bn + Cp = 0\)
• 点\((x_0, y_0, z_0)\)到平面\(Ax + By + Cz + D = 0\)的距离为\(d = \cfrac {|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}\)。
• 点\((x_0, y_0, z_0)\)到直线\(\cfrac {x - x_1}{m_1} = \cfrac {y - y_1}{n_1} = \cfrac {z - z_1}{p_1}\)的距离为\(d = \cfrac {|\{ x_1 - x_0, y_1 - y_0, z_1 - z_0 \} \times \{ m_1, n_1, p_1 \}|}{\sqrt{m_1^2 + n_1^2 + p_1^2}}\)。
• 两条不想交的直线的距离为\(d = \cfrac {|[\boldsymbol s_1 \boldsymbol s_2 \vec{AB}]|}{|\boldsymbol s_1 \times \boldsymbol s_2|}\)，其中\(A,B\)分别为两条直线上的一点。

3 曲面与曲线

曲线

曲线\(\Gamma\)的参数方程：\(\begin{cases} x = x(t) \\ y = y(t) \\ z = z(t) \\ \end{cases}\)，

• 切向量：\(\boldsymbol{\tau} = \{ x'(t_0), y'(t_0), z'(t_0\}\)
• 切线方程：\(\cfrac {x - x_0}{x'(t_0)} = \cfrac {y - y_0}{y'(t_0)} = \cfrac {z - z_0}{z'(t_0)}\)
• 法平面方程：\(x'(t_0)(x - x_0) + y'(t_0)(y - y_0) + z'(t_0)(z - z_0) = 0\)

曲线\(\Gamma\)的方程一般式：\(\begin{cases} F(x, y, z) = 0 \\ G(x, y, z) = 0 \\ \end{cases}\)，

• 切向量：\(\boldsymbol {\tau} = \boldsymbol{n_1} \times \boldsymbol{n_2}\)，其中\(\boldsymbol{n_1} = \{ F_x', F_y', F_z' \}, \boldsymbol{n_2} = \{ G_x', G_y', G_z' \}\)
• 切线方程：记\(\boldsymbol{n_1} \times \boldsymbol{n_2} = \{ A, B, C \}\)，\(\cfrac {x - x_0}A = \cfrac {y - y_0}B = \cfrac {z - z_0}C\)
• 法平面方程：\(A(x - x_0) + B(y - y_0) + C(z - z_0) = 0\)

曲面

曲面：\(F(x, y, z) = 0\)和其上一点\((x_0, y_0, z_0)\)

• 法向量：\(\boldsymbol{n} = \{F_x'(x_0, y_0, z_0), F_y'(x_0, y_0, z_0), F_z'(x_0, y_0, z_0)\}\)
• 切平面：\(F_x'(x_0, y_0, z_0)(x - x_0) + F_y'(x_0, y_0, z_0)(y - y_0) + F_z'(x_0, y_0, z_0)(z - z_0) = 0\)
• 法线方程：\(\cfrac {x - x_0}{F_x'(x_0, y_0, z_0)} = \cfrac {y - y_0}{F_y'(x_0, y_0, z_0)} = \cfrac {z - z_0}{F_z'(x_0, y_0, z_0)}\)

空间曲线在坐标面上的投影

设有空间曲线\(\Gamma:\begin{cases} F(x, y, z) = 0 \\ G(x, y, z) = 0 \\ \end{cases}\)，先消去\(z\)得\(\varphi (x, y) = 0\)，其投影的方程包含在\(\begin{cases} \varphi (x, y) = 0 \\ z = 0 \\ \end{cases}\)中。

（来自多元微分）

旋转曲面

• 定义：由一条平面曲线绕其平面上的一条直线旋转一周所成的曲面
• 设在\(xOy\)面上的曲线\(L\)：\(\begin{cases} f(x, y) = 0 \\ z = 0 \\ \end{cases}\)，则
  • 曲线\(L\)绕\(x\)轴旋转产生的曲面方程为\(f(x, \pm \sqrt{y^2 + z^2}) = 0\)
  • 曲线\(L\)绕\(y\)轴旋转产生的曲面方程为\(f(\pm \sqrt{x^2 + z^2}, y) = 0\)

柱面

• 定义：由一条直线（母线）沿定曲线（准线）平行移动形成的轨迹所成的曲面。

• 方程建立

  • 准线\(L\)：\(\begin{cases} F(x, y, z) = 0 \\ G(x, y, z) = 0 \\ \end{cases}\)，母线的方向向量为\(\{ m, n, p \}\)，

    在\(L\)上任取一点\((x_ 0, y_0, z_0)\)，则母线方程为\(\cfrac {x - x_0}m = \cfrac {y - y_0}n = \cfrac {z - z_0}p\)。

    联立方程\(\begin{cases} F(x_0, y_0, z_0) = 0 \\ G(x_0, y_0, z_0) = 0 \\ \cfrac {x - x_0}m = \cfrac {y - y_0}n = \cfrac {z - z_0}p \\ \end{cases}\)， 消去\(x_0, y_0, z_0\)，即可得到所求柱面方程。

  • 准线\(L\)：\(\begin{cases} x = x(t) \\ y = y(t) \\ z = z(t) \\ \end{cases}\)，母线的方向向量为\(\{ m, n, p \}\)，

    柱面方程为\(\begin{cases} x = x(t) + ms \\ y = y(t) + ns \\ z = z(t) + ps \\ \end{cases}\)，这里\(t, s\)均为参数。

  • 设柱面的准线是\(xOy\)平面上的曲线\(\begin{cases} f(x, y) = 0 \\ z = 0 \\ \end{cases}\)，母线平行与\(x\)轴，则柱面方程为\(f(x, y) = 0\)。

• 常用的柱面（这里的\(u, y\)为任意两个坐标轴）

  • 圆柱面 \(u^2 + v^2 = R^2\)
  • 椭圆柱面 \(\cfrac {u^2}{a^2} + \cfrac {v^2}{b^2} = 1\)
  • 抛物柱面 \(v^2 = 2pu\)

二次曲面

• 椭球面\(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} + \cfrac {z^2}{c^2} = 1\)
• 单叶双曲面 \(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 1\)
• 双叶双曲面 \(-\cfrac {x^2}{a^2} - \cfrac {y^2}{b^2} + \cfrac {z^2}{c^2} = 1\)
• 椭圆抛物面 \(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} = 2pz, \ (p \gt 0)\)
• 双曲抛物面 \(\cfrac {x^2}{a^2} - \cfrac {y^2}{b^2} = 2pz, \ (p \gt 0)\)
• 椭圆锥面（二次锥面） \(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 0\)