Equations of Planes | 平面方程

📚 Equations of Planes | 平面方程

In three-dimensional coordinate geometry, a plane is a flat, two-dimensional surface that extends infinitely. Being able to write and manipulate the equation of a plane is a core skill in A-Level Further Mathematics, especially for questions involving lines, angles and distances.

在三维坐标几何中，平面是一个平坦的、二维的、无限延伸的表面。能够写出并处理平面方程，是 A-Level 进阶数学中的核心技能，尤其是在涉及直线、夹角和距离的问题中尤为重要。

---

1. What is a Plane? | 什么是平面？

A plane can be uniquely determined by giving one point on the plane and two non-parallel direction vectors that lie in the plane. Alternatively, it can be determined by a single point and a normal vector (a vector perpendicular to the plane). These two approaches lead to different forms of the equation.

一个平面可以由平面上的一点以及两个位于平面内且互不平行的方向向量唯一确定。也可以由一个点和一个法向量（垂直于平面的向量）确定。这两种方法会得到不同形式的方程。

In this article we will study four equivalent representations: the vector equation with two direction vectors, the scalar-product form, the Cartesian (linear) form, and how to switch between them. We will also examine intersections, angles and distances.

本文将介绍四种等价表示：带两个方向向量的向量方程、点积形式、笛卡尔（线性）形式，以及它们之间的转换。我们还会讨论相交、夹角和距离问题。

---

2. Vector Equation with Two Direction Vectors | 带两个方向向量的向量方程

Let A be a point on the plane with position vector a. Let b and c be two non-parallel vectors parallel to the plane. Then any point on the plane with position vector r can be written as a linear combination of b and c, translated by a:

设 A 为平面上的点，其位置向量为 a。设 b 和 c 为两个与平面平行且互不平行的向量。那么平面上任意一点的位置向量 r 可以表示为 b 和 c 的线性组合，再加上 a 的平移：

r = a + λb + μc

where λ and μ are real parameters. Each choice of λ and μ picks out a different point on the plane.

其中 λ 和 μ 是实数参数。每取一组 λ 和 μ，就对应平面上的一个不同点。

It is essential that b and c are not parallel; otherwise the equation only describes a line.

必须确保 b 和 c 不平行；否则该方程只能描述一条直线。

---

3. Scalar Product Form | 点积形式

Let n be a normal vector to the plane. For any point r on the plane, the vector r − a lies inside the plane, so it is perpendicular to n. Hence:

设 n 为平面的法向量。对于平面上任意点 r，向量 r − a 位于平面内，因此它与 n 垂直，所以：

(r − a) · n = 0

Expanding the dot product gives:

展开点积可得：

r · n = a · n

If we write d = a · n, we obtain the compact scalar-product form of a plane:

若记 d = a · n，则得到平面的简洁点积形式：

r · n = d

Here d is a constant for a given plane. The normal vector n is the same regardless of which point A is used.

这里的 d 是给定平面的常数。无论选取哪个点 A，法向量 n 都是相同的。

---

4. Cartesian Equation of a Plane | 平面的笛卡尔方程

Write n = (A, B, C) and r = (x, y, z). Then the scalar-product form r · n = d becomes:

设 n = (A, B, C)，r = (x, y, z)，则点积形式 r · n = d 可化为：

A x + B y + C z = D

where D = d is a constant. This is the Cartesian equation of the plane. The vector (A, B, C) is the normal vector to the plane.

其中 D = d 为常数。这就是平面的笛卡尔方程，向量 (A, B, C) 为该平面的法向量。

Conversely, any equation of the form A x + B y + C z = D, where not all of A, B, C are zero, represents a plane with normal (A, B, C).

反过来，任何形如 A x + B y + C z = D 的方程（A、B、C 不全为零）都表示一个法向量为 (A, B, C) 的平面。

---

5. Converting Between Forms | 不同形式之间的转换

There are several useful conversions between the three main forms. The table below summarises the most common techniques.

三种主要形式之间有一些常用的转换技巧。下表总结了最常见的方法。

Desired form	Method
From point + two directions to Cartesian	Set n = b × c, then use r · n = a · n and expand.
From Cartesian to point + two directions	Find three non-collinear points on the plane, then set directions as the difference vectors.
From Cartesian to scalar product	Read the normal vector n = (A,B,C) and constant D, then r · n = D.

To convert from a point-and-two-directions form to Cartesian form, first compute the normal vector n = b × c. Then substitute the position vector a of the known point into r · n = d to find d.

若要从“一点加两个方向”形式转换为笛卡尔形式，先计算法向量 n = b × c，再将已知点的位置向量 a 代入 r · n = d 求出 d。

To convert from Cartesian to a vector form, find three non-collinear points on the plane by setting two variables to convenient values, then use the two difference vectors as direction vectors.

若要从笛卡尔形式转换为向量形式，可令两个变量取方便的值，求出平面上的三个不共线点，再用两条差向量作为方向向量。

---

6. Finding the Equation from Three Points | 由三个点求平面方程

Suppose three non-collinear points A, B and C have position vectors a, b and c. First form the direction vectors:

设三个不共线点 A、B、C 的位置向量分别为 a、b、c。首先构造方向向量：

u = b − a, v = c − a

Then the normal vector is given by the cross product of u and v:

则法向量由 u 和 v 的叉积给出：

n = u × v = (b − a) × (c − a)

Finally the equation of the plane is r · n = a · n, which can be written in Cartesian form.

最终平面方程为 r · n = a · n，可进一步写成笛卡尔形式。

This method is robust because the cross product automatically gives a vector perpendicular to both direction vectors, and hence perpendicular to the plane.

这种方法很稳健，因为叉积自动给出与两个方向向量都垂直的向量，因此也就是平面的法向量。

---

7. Intersection of a Line and a Plane | 直线与平面相交

A line in 3D can be written in vector form as r = p + t q, where p is the position vector of a point on the line, q is the direction vector, and t is a parameter. To find where this line meets a plane r · n = d, substitute the line equation into the plane equation:

三维空间中的直线可写为向量形式 r = p + t q，其中 p 为直线上一点的向量，q 为方向向量，t 为参数。要求该直线与平面 r · n = d 的交点，将直线方程代入平面方程：

(p + t q) · n = d

Expanding and solving for t gives:

展开并解出 t：

t = (d − p · n) / (q · n)

provided that q · n ≠ 0. If q · n = 0, the line is parallel to the plane. In that case, if p · n = d, the line lies entirely in the plane; otherwise there is no intersection.

前提是 q · n ≠ 0。若 q · n = 0，则直线与平面平行。此时若 p · n = d，则直线完全位于平面内；否则直线与平面没有交点。

Once t is found, substitute it back into the line equation to obtain the point of intersection.

求出 t 后，代回直线方程即可得到交点坐标。

---

8. 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 normals n₁ and n₂, then the cosine of the angle θ satisfies:

两个平面之间的夹角定义为它们法向量之间的锐角。若两平面的法向量分别为 n₁ 和 n₂，则夹角 θ 的余弦满足：

cos θ = |n₁ · n₂| / (|n₁||n₂|)

The absolute value ensures that we obtain the acute angle, not the obtuse one. If the dot product is zero, the planes are perpendicular.

取绝对值是为了得到锐角而不是钝角。若点积为零，则两平面互相垂直。

In Cartesian form, if the two planes are A₁x + B₁y + C₁z = D₁ and A₂x + B₂y + C₂z = D₂, then the angle

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）