1. 向量及其运算
1. 向量的数量积 (点乘积或内积)
向量 \(\va{a} = \left\{ {{a}_{1},{a}_{2},{a}_{3}}\right\}\) 与 \(\va{b} = \left\{ {{b}_{1},{b}_{2},{b}_{3}}\right\}\) 的数量积是一个数 \(\left| \va{a}\right| \vdot \left| \va{b}\right| \cos \left( {\va{a},\va{b}}\right)\) ,(且 \(0 \leq \left( {\va{a},\va{b}}\right)\) \(\leq \pi\) ),记作 \(\va{a} \vdot \va{b}\) . 若向量 \(\va{a}\) 或 \(\va{b}\) 为零向量时,则定义 \(\va{a} \vdot \va{b} = 0\) ,数量积 \(\va{a} \vdot \va{b}\) 的坐标表示式为\[\va{a} \vdot \va{b} = {a}_{1}{b}_{1} + {a}_{2}{b}_{2} + {a}_{3}{b}_{3}\]两个向量 \(\va{a},\va{b}\) 垂直 (或称正交),记作 \(\va{a} \bot \va{b}\) ,特别地,规定零向量与任一向量垂直
数量积有以下基本性质:
(1) \(\va{a} \vdot \va{b} = \va{b} \vdot \va{a}\)
(2) \(\left( {\lambda \va{a}}\right) \vdot \va{b} = \lambda \left( {\va{a} \vdot \va{b}}\right)\)
(3) \(\left( {\va{a} + \va{b}}\right) \vdot \va{c} = \va{a} \vdot \va{c} + \va{b} \vdot \va{c}\)
(4) \(\va{a} \bot \va{b}\) 的充分必要条件是 \(\va{a} \vdot \va{b} = 0\)
2. 向量的向量积 (叉乘积或外积)
两个向量 \(\va{a}\) 和 \(\va{b}\) 的向量积是一个向量 \(\va{c}\) ,记为 \(\va{a} \times \va{b}\) ,即 \(\va{c} = \va{a} \times \va{b};\va{c}\) 的模等于 \(\left| a\right| \left| b\right| \sin \left( {\va{a},\va{b}}\right) ,\va{c}\) 的方向垂直于 \(\va{a}\) 与 \(\va{b}\) 所决定的平面,且 \(\va{a},\va{b},\va{c}\) 顺次构成右手系. 若向量 \(\va{a}\) 或 \(\va{b}\) 为零向量时,则定义 \(\va{a} \times \va{b} = \va{0}\) ,向量积 \(\va{a} \times \va{b}\) 坐标表示式为\[\va{a} \times \va{b} = \left| \begin{array}{rrr} \vu{i} & \vu{j} & \vu{k} \\ {a}_{1} & {a}_{2} & {a}_{3} \\ {b}_{1} & {b}_{2} & {b}_{3} \end{array}\right| = \left\{ {\left| \begin{array}{ll} {a}_{2} & {a}_{3} \\ {b}_{2} & {b}_{3} \end{array}\right| , - \left| \begin{array}{ll} {a}_{1} & {a}_{3} \\ {b}_{1} & {b}_{3} \end{array}\right| ,\left| \begin{array}{ll} {a}_{1} & {a}_{2} \\ {b}_{1} & {b}_{2} \end{array}\right| }\right\}\]向量积有以下的性质:
(1) \(\va{a} \times \va{b} = - \va{b} \times \va{a}\)
(2) \(\left( {\lambda \va{a}}\right) \times \va{b} = \lambda \left( {\va{a} \times \va{b}}\right)\)
(3) \(\left( {\va{a} + \va{b}}\right) \times \va{c} = \va{a} \times \va{c} + \va{b} \times \va{c}\)
(4) \(\va{a}//\va{b}\) 的充分必要条件是 \(\va{a} \times \va{b} = \va{0}\)
3. 向量的混合积
设 \(\va{a} = \left\{ {{a}_{1},{a}_{2},{a}_{3}}\right\} ,\va{b} = \left\{ {{b}_{1},{b}_{2},{b}_{3}}\right\} ,\va{c} = \left\{ {{c}_{1},{c}_{2},{c}_{3}}\right\}\) ,则称乘积 \(\left( {\va{a} \times \va{b}}\right) \vdot \va{c}\) 为向量 \(\va{a},\va{b},\va{c}\) 的混合积,记为 \(\left\lbrack{\va{a},\va{b},\va{c}}\right\rbrack\)
混合积是一数量,其几何意义为: 混合积的绝对值等于以 \(\va{a}\text{ 、 }\va{b}\text{ 、 }\va{c}\) 为相邻三条棱的平行六面体的体积. 因此,向量 \(\va{a}\text{ 、 }\va{b}\text{ 、 }\va{c}\) 共面的充分必要条件是 \(\left( {\va{a} \times \va{b}}\right) \vdot \va{c} = 0\)
混合积 \(\left( {\va{a} \times \va{b}}\right) \vdot \va{c}\) 的坐标表达式为 \(\left( {\va{a} \times \va{b}}\right) \vdot \va{c} = \left| \begin{array}{lll} {a}_{1} & {a}_{2} & {a}_{3} \\ {b}_{1} & {b}_{2} & {b}_{3} \\ {c}_{1} & {c}_{2} & {c}_{3} \end{array}\right|\)
且 \(\left( {\va{a} \times \va{b}}\right) \vdot \va{c} = \left( {\va{b} \times \va{c}}\right) \vdot \va{a} = \left( {\va{c} \times \va{a}}\right) \vdot \va{b}\)
2. 空间的平面和直线
1. 平面及其方程
法向量 与平面垂直的任意非零向量, 称为该平面的法向量.
(1)点法式方程 设平面过点 \({M}_{0}\left( {{x}_{0},{y}_{0},{z}_{0}}\right)\) ,其法向量为 \(\va{n} = \{ A,B,C\}\) 则此平面方程为\[A\left( {x - {x}_{0}}\right) + B\left( {y - {y}_{0}}\right) + C\left( {z - {z}_{0}}\right) = 0\](2)截距式方程 设 \(a,b,c\) 分别为平面在 \(x\text{ 、 }y\text{ 、 }z\) 轴上的截距,则此平面的方程为\[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\](3)三点式方程 设平面过不共线的三点 \(A\left( {{x}_{1},{y}_{1},{z}_{1}}\right) ,B\left( {{x}_{2},{y}_{2},{z}_{2}}\right) ,C\left( {{x}_{3},{y}_{3},{z}_{3}}\right)\) ,则此平面方程为\[\left| \begin{matrix} x - {x}_{1} & y - {y}_{1} & z - {z}_{1} \\ {x}_{2} - {x}_{1} & {y}_{2} - {y}_{1} & {z}_{2} - {z}_{1} \\ {x}_{3} - {x}_{1} & {y}_{3} - {y}_{1} & {z}_{3} - {z}_{1} \end{matrix}\right| = 0\](4)一般式方程 平面的一般式方程是三元一次方程\[{Ax} + {By} + {Cz} + D = 0\]其中 \(A,B,C\) 不同时为零.
2. 空间直线及其方程
方向向量 与直线平行的非零向量, 称为该直线的方向向量.
(1)对称式方程(又称点向式或标准式方程) 过点 \({M}_{0}\left( {{x}_{0},{y}_{0},{z}_{0}}\right)\) ,方向向量为 \(\va{s} = \{ l,m,n\}\) 的直线的标准式方程为\[\frac{x - {x}_{0}}{l} = \frac{y - {y}_{0}}{m} = \frac{z - {z}_{0}}{n}\](2)参数方程 由标准式方程\[\frac{x - {x}_{0}}{l} = \frac{y - {y}_{0}}{m} = \frac{z - {z}_{0}}{n} = t\]易得直线的参数方程\[\left\{ \begin{array}{l} x = {x}_{0} + {lt} \\ y = {y}_{0} + {mt}\;\quad\left( {t\text{ 为参数 }}\right) \\ z = {z}_{0} + {nt} \end{array}\right.\](3) 两点式方程 过点 \({M}_{1}\left( {{x}_{1},{y}_{1},{z}_{1}}\right)\) 和 \({M}_{2}\left( {{x}_{2},{y}_{2},{z}_{2}}\right)\) 的直线方程为\[\frac{x - {x}_{1}}{{x}_{2} - {x}_{1}} = \frac{y - {y}_{1}}{{y}_{2} - {y}_{1}} = \frac{z - {z}_{1}}{{z}_{2} - {z}_{1}}\](4)一般式方程 直线的一般式方程为三元一次方程组\[\left\{ \begin{array}{l} {A}_{1}x + {B}_{1}y + {C}_{1}z + {D}_{1} = 0 \\ {A}_{2}x + {B}_{2}y + {C}_{2}z + {D}_{2} = 0 \end{array}\right.\]其中每一个三元一次方程都表示一个平面.
3. 直线、平面之间的相对位置关系
设平面 \({\pi }_{1} : {A}_{1}x + {B}_{1}y + {C}_{1}z + {D}_{1} = 0\;{\pi }_{2} : {A}_{2}x + {B}_{2}y + {C}_{2}z + {D}_{2} = 0\) 它们的法向量分别为 \({\va{n}}_{1} = \left\{ {{A}_{1},{B}_{1},{C}_{1}}\right\} ,{\va{n}}_{2} = \left\{ {{A}_{2},{B}_{2},{C}_{2}}\right\}\)
直线 \(\displaystyle {L}_{1} : \frac{x - {x}_{1}}{{l}_{1}} = \frac{y - {y}_{1}}{{m}_{1}} = \frac{z - {z}_{1}}{{n}_{1}}\;{L}_{2} : \frac{x - {x}_{2}}{{l}_{2}} = \frac{y - {y}_{2}}{{m}_{2}} = \frac{z - {z}_{2}}{{n}_{2}}\) 它们的方向向量分别为 \({\va{s}}_{1} = \left\{ {{l}_{1},{m}_{1},{n}_{1}}\right\} ,{\va{s}}_{2} = \left\{ {{l}_{2},{m}_{2},{n}_{2}}\right\}\)
(1) 夹角 平面 \({\pi }_{1}\) 与平面 \({\pi }_{2}\) 间的夹角 \(\theta\) 定义为法向量 \({\va{n}}_{1}\) 与 \({\va{n}}_{2}\) 间的夹角,即\[\cos \theta = \frac{\left| {\va{n}}_{1} \vdot {\va{n}}_{2}\right| }{\left| {\va{n}}_{1}\right| \vdot \left| {\va{n}}_{2}\right| } = \frac{\left| {A}_{1}{A}_{2} + {B}_{1}{B}_{2} + {C}_{1}{C}_{2}\right| }{\sqrt{{A}_{1}^{2} + {B}_{1}^{2} + {C}_{1}^{2}} \vdot \sqrt{{A}_{2}^{2} + {B}_{2}^{2} + {C}_{2}^{2}}}\]直线 \({L}_{1}\) 与直线 \({L}_{2}\) 间的夹角 \(\theta\) 定义为方向向量 \({s}_{1}\) 与 \({s}_{2}\) 间的夹角,即\[\cos \theta = \frac{\left| {s}_{1} \vdot {s}_{2}\right| }{\left| {s}_{1}\right| \vdot \left| {s}_{2}\right| } = \frac{\left| {l}_{1}{l}_{2} + {m}_{1}{m}_{2} + {n}_{1}{n}_{2}\right| }{\sqrt{{l}_{1}^{2} + {m}_{1}^{2} + {n}_{1}^{2}} \vdot \sqrt{{l}_{2}^{2} + {m}_{2}^{2} + {n}_{2}^{2}}}\]直线 \({L}_{1}\) 与平面 \({\pi }_{1}\) 间的夹角 \(\theta\) 定义为直线 \({L}_{1}\) 和它在平面 \({\pi }_{1}\) 上的投影所成的两邻角中的锐角, 即\[\sin \theta = \frac{\left| {\va{n}}_{1} \vdot {\va{s}}_{1}\right| }{\left| {\va{n}}_{1}\right| \vdot \left| {\va{s}}_{1}\right| } = \frac{\left| {A}_{1}{l}_{1} + {B}_{1}{m}_{1} + {C}_{1}{n}_{1}\right| }{\sqrt{{A}_{1}^{2} + {B}_{1}^{2} + {C}_{1}^{2}} \vdot \sqrt{{l}_{1}^{2} + {m}_{1}^{2} + {n}_{1}^{2}}}\](2)平行的条件 平面 \({\pi }_{1}\) 与 \({\pi }_{2}\) 平行的充分必要条件是 \(\displaystyle\frac{{A}_{1}}{{A}_{2}} = \frac{{B}_{1}}{{B}_{2}} = \frac{{C}_{1}}{{C}_{2}}\)
直线 \({L}_{1}\) 与 \({L}_{2}\) 平行的充分必要条件是 \(\displaystyle\frac{{l}_{1}}{{l}_{2}} = \frac{{m}_{1}}{{m}_{2}} = \frac{{n}_{1}}{{n}_{2}}\)
直线 \({L}_{1}\) 与平面 \({\pi }_{1}\) 平行的充分必要条件是 \({l}_{1}{A}_{1} + {m}_{1}{B}_{1} + {n}_{1}{C}_{1} = 0\)
(3)垂直的条件 平面 \({\pi }_{1}\) 与 \({\pi }_{2}\) 垂直的充分必要条件是 \({A}_{1}{A}_{2} + {B}_{1}{B}_{2} + {C}_{1}{C}_{2} = 0\)
直线 \({L}_{1}\) 与 \({L}_{2}\) 垂直的充分必要条件是 \({l}_{1}{l}_{2} + {m}_{1}{m}_{2} + {n}_{1}{n}_{2} = 0\)
直线 \({L}_{1}\) 垂直于平面 \({\pi }_{1}\) 的充分必要条件是 \(\displaystyle\frac{{l}_{1}}{{A}_{1}} = \frac{{m}_{1}}{{B}_{1}} = \frac{{n}_{1}}{{C}_{1}}\)
4. 距离公式
(1)点到平面的距离 点 \({M}_{0}\left( {{x}_{0},{y}_{0},{z}_{0}}\right)\) 到平面 \({Ax} + {By} + {Cz} + D = 0\) 的距离为 \(\displaystyle d = \frac{\left| A{x}_{0} + B{y}_{0} + C{z}_{0} + D\right| }{\sqrt{{A}^{2} + {B}^{2} + {C}^{2}}}\)
(2)点到直线的距离 点 \({P}_{1}\left( {{x}_{1},{y}_{1},{z}_{1}}\right)\) 到直线 \(\displaystyle\frac{x - {x}_{0}}{l} = \frac{y - {y}_{0}}{m} = \frac{z - {z}_{0}}{n}\) 的距离为 \(\displaystyle d = \frac{\left| \overrightarrow{{M}_{0}{P}_{1}} \times s\right| }{\left| s\right| }\) ,其中,\[{M}_{0}\left( {{x}_{0},{y}_{0},{z}_{0}}\right) ,\;\va{s} = \{ l,m,n\}\](3)两直线共面的条件 设有两直线 \(\displaystyle {L}_{1} : \frac{x - {x}_{1}}{{l}_{1}} = \frac{y - {y}_{1}}{{m}_{1}} = \frac{z - {z}_{1}}{{n}_{1}},\;{L}_{2} : \frac{x - {x}_{2}}{{l}_{2}} = \frac{y - {y}_{2}}{{m}_{2}} = \frac{z - {z}_{2}}{{n}_{2}}\)
共面的条件为 \(\overrightarrow{{P}_{1}{P}_{2}} \vdot \left( {\va{a} \times \va{b}}\right) = 0\) ,其中\[{P}_{1}\left( {{x}_{1},{y}_{1},{z}_{1}}\right) ,\;{P}_{2}\left( {{x}_{2},{y}_{2},{z}_{2}}\right) ,\;\va{a} = \left\{ {{l}_{1},{m}_{1},{n}_{1}}\right\} ,\;\va{b} = \left\{ {{l}_{2},{m}_{2},{n}_{2}}\right\}\](4)两直线间的距离 两异面直线 \({L}_{1},{L}_{2}\) 的距离为 \(\displaystyle d = \frac{\left| \overrightarrow{{P}_{1}{P}_{2}} \vdot \left( \va{a} \times \va{b}\right) \right| }{\left| \va{a} \times \va{b}\right| }\)
3. 空间曲面与空间曲线
1. 空间曲面方程
(1)一般方程 \(F\left( {x,y,z}\right) = 0\)
(2)显式方程 \(z = f\left( {x,y}\right)\)
(3)参数方程 \(\displaystyle\left\{ \begin{array}{l} x = x\left( {u,v}\right) \\ y = y\left( {u,v}\right) \left( {u,v}\right) \in D, \\ z = z\left( {u,v}\right) \end{array}\right.\) 其中 \(D\) 为 \({uv}\) 平面上某一区域.
2. 旋转曲面方程
设\(C : f\left( {y,z}\right) = 0\) 为 \({yOz}\) 平面上的曲线,则
(1)\(C\) 绕 \(z\) 轴旋转所得的曲面为 \(f\left( {\pm \sqrt{{x}^{2} + {y}^{2}},z}\right) = 0\)
(2)\(C\) 绕 \(y\) 轴旋转所得的曲面为 \(f\left( {y, \pm \sqrt{{x}^{2} + {z}^{2}}}\right) = 0\)
旋转曲面主要由母线和旋转轴确定.
求旋转曲面方程时, 平面曲线绕某坐标轴旋转, 则该坐标轴对应的变量不变, 而曲线方程中另一变量改写成该变量与第三变量平方和的正负平方根,例如: \(\displaystyle L\left\{ \begin{array}{l} f\left( {x,y}\right) = 0 \\ z = 0 \end{array}\right.\) . 曲线 \(L\) 绕 \(x\) 轴旋转所形成的旋转曲面的方程为 \(f\left( {x, \pm \sqrt{{y}^{2} + {z}^{2}}}\right) = 0\)
3. 柱面方程
(1)母线平行于 \(z\) 轴的柱面方程为 \(F\left( {x,y}\right) = 0\)
(2)母线平行于 \(x\) 轴的柱面方程为 \(G\left( {y,z}\right) = 0\)
(3)母线平行于 \(y\) 轴的柱面方程为 \(H\left( {x,z}\right) = 0\)
当曲面方程中缺少一个变量时,则曲面为柱面. 如 \(F\left( {x,y}\right) = 0\) ,变量 \(z\) 未出现,该曲面表示由准线 \(\displaystyle\left\{ \begin{array}{l} F\left( {x,y}\right) = 0 \\ z = 0 \end{array}\right.\) 生成,母线平行于 \(z\) 轴的柱面.
柱面方程必须注意准线与母线两个要素.
4.六个典型曲面
(1)椭圆锥面\(\displaystyle\quad\frac{x^2}{a^2}+\frac{y^2}{b^2}=z^2\)
(2)椭球面\(\displaystyle\quad\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)
(3)单叶双曲面\(\displaystyle\quad\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)
(4)双叶双曲面\(\displaystyle\quad\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)
(5)椭圆抛物面\(\displaystyle\quad\frac{x^2}{a^2}+\frac{y^2}{b^2}=z\)
(6)双曲抛物面(马鞍面)\(\displaystyle\quad\frac{x^2}{a^2}-\frac{y^2}{b^2}=z\)