This is a beta course, so its structure, chapters, and examples may continue to change.

Analytic Geometry of Space

Points and Coordinates in Space

A point in space is determined by an ordered triple $(x, y, z)$ . The distance between $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$ is

|P_1P_2| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

Planes in Space

General Equation

General equation of a plane

$Ax + By + Cz + D = 0$

where $\vec{n} = (A, B, C)$ is a normal vector of the plane.

$\vec{n}$ is perpendicular to every vector lying in the plane.
$A, B, C$ cannot all be zero; multiplying the equation by a nonzero constant does not change the plane.

Point-Normal Equation

Point-normal form

The plane through $P_0(x_0, y_0, z_0)$ with normal $\vec{n} = (A, B, C)$ is

$A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$

Intercept Equation

Intercept form

$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

where $a, b, c$ are the intercepts on the $x, y, z$ axes.

Lines in Space

Parametric Equation

Parametric equation of a line

The line through $P_0(x_0, y_0, z_0)$ with direction vector $\vec{s} = (l, m, n)$ :

\begin{cases} x = x_0 + lt \\ y = y_0 + mt \\ z = z_0 + nt \end{cases}

where $t$ is the parameter.

$t > 0$ : move along the positive direction
$t < 0$ : move along the negative direction
$t = 0$ : at the starting point

Symmetric Equation

Symmetric equation of a line

$\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$

Two-Point Equation

Two-point form

The line through $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ :

$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$

with direction vector $\vec{P_1P_2} = (x_2-x_1, y_2-y_1, z_2-z_1)$ .

General Equation

A line can also be written as the intersection of two planes:

General form of a line

\begin{cases} A_1x + B_1y + C_1z + D_1 = 0 \\ A_2x + B_2y + C_2z + D_2 = 0 \end{cases}

Positional Relationships

Two Lines

Parallel: direction vectors are proportional.
Intersecting: they share a common point.
Skew: neither parallel nor intersecting.

Two lines are coplanar iff the scalar triple product of their two direction vectors and the connecting vector of a point on each line is zero.

Lines and Planes

Line parallel to plane: the direction vector of the line is perpendicular to the normal of the plane, and the line does not lie in the plane.
Line lies in the plane: in addition, a point of the line lies in the plane.
Line perpendicular to plane: the direction vector is parallel to the normal vector.

Two Planes

Parallel: normals are proportional.
Perpendicular: normals are perpendicular.

Distance and Angle Formulas

Distance from a Point to a Plane

Point-to-plane distance

The distance from $P_0(x_0,y_0,z_0)$ to $Ax+By+Cz+D=0$ is

$d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$

Distance from a Point to a Line

Point-to-line distance

For a line through $P_0$ with direction $\vec{s}$ , the distance from point $P$ is

$d = \frac{|\vec{P_0P} \times \vec{s}|}{|\vec{s}|}$

Angle Between Two Lines

Angle between lines

With direction vectors $\vec{s}_1, \vec{s}_2$ ,

$\cos\theta = \frac{|\vec{s}_1 \cdot \vec{s}_2|}{|\vec{s}_1||\vec{s}_2|}$

Angle Between Two Planes

Angle between planes

With normals $\vec{n}_1, \vec{n}_2$ ,

$\cos\theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$

Angle Between a Line and a Plane

Line-plane angle

With line direction $\vec{s}$ and plane normal $\vec{n}$ ,

$\sin\theta = \frac{|\vec{s} \cdot \vec{n}|}{|\vec{s}||\vec{n}|}$

Tip: the line–plane angle uses $\sin$ (the angle between the line and its projection in the plane), while line–line and plane–plane angles use $\cos$ . The absolute value keeps the acute angle.

Summary

Symbols Used in This Article

Symbol	Type	Reading/Explanation	Meaning in This Article
$\vec{n}$	vector	n	The normal vector of a plane
$\vec{s}$	vector	s	The direction vector of a line
$t$	parameter	t	The parameter of a line
$d$	distance	d	A distance
$\theta$	Greek letter	theta	An angle

English–Chinese Glossary

English term	Chinese term	Phonetic	Explanation
normal vector	法向量	/ˈnɔːml ˈvɛktər/	A vector perpendicular to a plane
direction vector	方向向量	/dɪˈrekʃən ˈvɛktər/	A vector along a line
parametric equation	参数方程	/ˌpærəˈmetrɪk ɪˈkweɪʒən/	A representation with a parameter
symmetric equation	对称式方程	/sɪˈmetrɪk ɪˈkweɪʒən/	The symmetric form of a line
skew lines	异面直线	/skjuː laɪnz/	Lines neither parallel nor intersecting
point-to-plane distance	点到平面的距离	/pɔɪnt tuː pleɪn ˈdɪstəns/	Perpendicular distance from a point to a plane
intercept	截距	/ˌɪntəˈsept/	Where a plane meets an axis

Products of Vectors

Learn the dot product, cross product, and scalar triple product of vectors, together with their geometric meaning and applications.

Quadric Surfaces

Learn the general equation, classification, and standard forms of quadric surfaces—spheres, ellipsoids, paraboloids, hyperboloids, and cones.