Line and Plane | 直线与平面

📚 Line and Plane | 直线与平面

In three-dimensional geometry, lines and planes are fundamental objects. Mastering their vector and Cartesian equations, intersections, distances, and angles is essential for success in IB Mathematics: Analysis and Approaches HL. This article provides a comprehensive review of key concepts and techniques, supported by clear formulas and examples.

在三维几何中，直线与平面是最基本的几何对象。熟练掌握它们的向量方程与一般方程、交点、距离和角度，是学好 IB 数学分析与方法（HL）的关键。本文将系统梳理相关核心概念与技巧，配以清晰的公式和实例。

1. Vector and Parametric Equations of a Line | 直线的向量方程与参数方程

A line in 3D space is uniquely determined by a point on the line and a direction vector. If the position vector of a known point is a and a direction vector is b, then every point on the line can be expressed by the vector equation

三维空间中的直线由直线上一点和一个方向向量唯一确定。如果已知点的位置向量为 a，方向向量为 b，那么直线上每一点都可用向量方程表示为：

r = a + t b , t ∈ ℝ

Here r is the position vector of a general point on the line, and the scalar parameter t varies over all real numbers. By writing a = (x₀, y₀, z₀) and b = (l, m, n), we obtain the parametric equations:

其中 r 为直线上任意点的位置向量，标量参数 t 取遍所有实数。设 a = (x₀, y₀, z₀)，b = (l, m, n)，即得参数方程：

x = x₀ + t l , y = y₀ + t m , z = z₀ + t n , t ∈ ℝ

These forms are extremely useful for finding intersections, distances, and for converting to other representations.

这些形式在求交点、距离以及转换为其他表示时非常有用。

2. Symmetric and Cartesian Forms of a Line | 直线的对称方程与一般形式

When none of the direction components l, m, n is zero, we can eliminate the parameter t from the parametric equations to obtain the symmetric form:

当方向向量的分量 l, m, n 均不为零时，我们可以从参数方程中消去参数 t，得到对称方程：

(x – x₀)/l = (y – y₀)/m = (z – z₀)/n

If one or two components are zero, the symmetric form cannot be written in the standard way. For instance, if l = 0 but m, n ≠ 0, the line is perpendicular to the x-axis and its x-coordinate remains constant. The representation then becomes x = x₀ and (y – y₀)/m = (z – z₀)/n. These are often called Cartesian equations of the line.

若有一或两个分量为零，则不能写成标准的对称式。例如，若 l = 0 而 m, n ≠ 0，则直线与 x 轴垂直，其 x 坐标保持不变。此时表示为 x = x₀，且 (y – y₀)/m = (z – z₀)/n。这通常称为直线的一般方程（笛卡尔形式）。

3. Equations of a Plane | 平面的方程

A plane in 3D can be defined by a point and a normal vector n. If a is the position vector of a point on the plane, the vector equation is

三维空间中的平面可由一点和一个法向量 n 定义。若 a 为平面上一点的位置向量，则向量方程为：

r · n = a · n

Alternatively, using two non-parallel direction vectors u and v lying in the plane, a parametric form is r = a + s u + t v, where s, t ∈ ℝ. Expanding the dot product with n = (a, b, c) and letting d = a · n gives the Cartesian equation:

也可用平面内两个不共线的方向向量 u 和 v 给出参数形式 r = a + s u + t v，其中 s, t ∈ ℝ。将点积展开，设 n = (a, b, c) 并令 d = a · n，就得到笛卡尔方程（一般式）：

ax + by + cz = d

The coefficients a, b, c are the components of a normal vector. To find a plane through three non-collinear points, one can form two direction vectors and take their cross product to obtain n.

其中系数 a, b, c 是法向量的分量。若要通过三个不共线点求平面，可构造两个方向向量并取其叉乘得到 n。

4. Intersection of a Line and a Plane | 直线与平面的交点

To find the intersection of a line r = a + t b and a plane ax + by + cz = d, substitute the parametric coordinates into the plane equation and solve for t. If a unique t exists, the line meets the plane at a single point. If the equation reduces to a contradiction (e.g., 0 = 5), the line is parallel to the plane and does not intersect. If it reduces to a true statement (0 = 0) for all t, the line lies entirely in the plane.

求直线 r = a + t b 与平面 ax + by + cz = d 的交点时，将参数坐标代入平面方程并解出 t。若存在唯一 t，则直线与平面交于一点。若方程化简为矛盾式（如 0=5），则直线平行于平面且不相交。若对所有 t 恒成立（0=0），则直线完全位于平面内。

The condition for parallelism is b · n = 0. If also a · n = d, the line lies in the plane.

平行的条件是 b · n = 0。若同时还满足 a · n = d，则直线在平面内。

5. Angle Between a Line and a Plane | 直线与平面的夹角

The angle θ between a line and a plane is defined as the acute angle between the line and its orthogonal projection onto the plane. It is related to the normal vector n of the plane and the direction vector b of the line by

直线与平面的夹角 θ 定义为直线与其在平面上正投影之间的锐角。它与平面的法向量 n 及直线的方向向量 b 的关系为：

sin θ = |b · n| / (|b| |n|)

Note that if b · n = 0, the line is parallel to the plane (θ = 0) or lies in it. The formula uses sine because the direction vector and the normal vector are perpendicular when the line is parallel to the plane.

注意，若 b · n = 0，则直线平行于平面（θ = 0）或在平面内。公式使用正弦，是因为当直线平行于平面时，方向向量与法向量垂直，夹角为90°。

6. Distance from a Point to a Plane | 点到平面的距离

The shortest distance from a point P with position vector p to the plane ax + by + cz = d is given by

点 P（位置向量为 p）到平面 ax + by + cz = d 的最短距离为：

d = |a x₀ + b y₀ + c z₀ – d| / √(a² + b² + c²)

In vector form, if a is a point on the plane and n is a normal vector, the distance is

用向量形式表示，若 a 为平面上一点，n 为法向量，则距离为：

d = |(p – a) · n| / |n|

This expression simply projects the vector from the plane to the point onto the unit normal direction.

该表达式本质上是将平面到点的向量投影到单位法向量方向上。

7. Distance from a Point to a Line in 3D | 点到直线的距离

To find the perpendicular distance from a point P (p) to a line given by r = a + t b, we use the cross product:

求点 P（p）到直线 r = a + t b 的垂直距离时，可利用叉乘：

d = |(p – a) × b| / |b|

This works because the area of the parallelogram formed by p – a and b is |(p – a) × b|, and dividing by the base length |b| gives the height (the distance).

其原理在于，以 p – a 和 b 为邻边的平行四边形的面积为 |(p – a) × b|，除以底边长 |b| 就得到高（即距离）。

8. Relationship Between Two Lines | 两条直线的关系

Two lines in 3D can be intersecting, parallel (including coincident), or skew (non-parallel and non-intersecting). Let the lines be r₁ = a₁ + t b₁ and r₂ = a₂ + s b₂. The direction vectors reveal parallelism: if b₁ and b₂ are proportional, the lines are parallel. To check for intersection, solve a₁ + t b₁ = a₂ + s b₂ for t and s. If a solution exists, the lines intersect at a point. If the direction vectors are not proportional and no solution exists, the lines are skew. A compact test for coplanarity (and hence non-skew) is

三维空间中的两条直线可能相交、平行（包括重合）或异面（不平行也不相交）。设两直线为 r₁ = a₁ + t b₁ 和 r₂ = a₂ + s b₂。方向向量可判断平行：若 b₁ 与 b₂ 成比例，则两直线平行。要判断是否相交，可解 a₁ + t b₁ = a₂ + s b₂ 求 t 和 s。若有解，则两直线交于一点。若方向向量不成比例且方程组无解，则它们为异面直线。判断共面（即不异面）的简洁条件是：

(a₂ – a₁) · (b₁ × b₂) = 0

If this scalar triple product is zero, the lines lie in a common plane and are either intersecting or parallel; otherwise they are skew.

若该标量三重积为零，则两直线共面，可能相交或平行；否则为异面直线。

9. Intersection of Two Planes – The Line of Intersection | 两平面的交线

When two planes are not parallel, they intersect in a straight line. The direction vector of the line of intersection is parallel to the cross product of their normal vectors, n₁ × n₂. To find a specific point on the line, we can set one coordinate to a convenient value (often 0) and solve the two plane equations simultaneously for the other two coordinates. If the normal vectors are parallel, the planes are either parallel or coincident, and there is no unique line of intersection.

当两平面不平行时，它们相交于一条直线。交线的方向向量平行于两法向量的叉乘 n₁ × n₂。要求交线上的一点，可将某一坐标设为方便的值（通常为0），然后联立两平面方程解出另外两个坐标。若法向量平行，则两平面平行或重合，不存在唯一的交线。

10. Angle Between Two Planes | 两平面之间的夹角

The angle between two planes is defined as the acute angle between their normal vectors. If the planes have normal vectors n₁ and n₂, the angle θ is given by

两平面的夹角定义为其法向量夹角中的锐角。若两平面法向量分别为 n₁ 和 n₂，则夹角 θ 满足：

cos θ = |n₁ · n₂| / (|n₁| |n₂|)

The absolute value ensures we obtain the acute angle (0 ≤ θ ≤ π/2). This is particularly useful when finding angles between faces of polyhedra.

取绝对值是为确保得到锐角（0 ≤ θ ≤ π/2）。

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Line and Plane | 直线与平面