📚 Equation of a Plane | 平面方程
A plane in three-dimensional space is a flat, two-dimensional surface that extends infinitely. In IB Mathematics, particularly in the Analysis and Approaches (AA) and Applications and Interpretation (AI) courses at Higher Level, understanding the equation of a plane is vital for solving problems involving vectors, geometry, and intersections. A plane can be uniquely determined by a point and a normal vector, or by a point and two non-parallel direction vectors that lie on the plane. Mastering the different forms of the plane equation enables students to switch seamlessly between geometric and algebraic approaches.
三维空间中的平面是一个无限延伸的平坦二维曲面。在IB数学中，尤其是高级水平的分析与方法(AA)和应用与解释(AI)课程，理解平面方程对于解决向量、几何与相交问题至关重要。一个平面可以由一个点和一个法向量唯一确定，也可以由平面上的一个点以及两个不共线的方向向量确定。熟练掌握平面方程的各种形式，能让学生自如地在几何与代数方法之间切换。
1. Introduction to Planes in 3D Space | 三维空间中的平面简介
In 3D coordinate geometry, a plane is defined as the set of all points (x, y, z) satisfying a linear equation. Conceptually, it is the 3D analogue of a line in 2D. Unlike a line, which has one degree of freedom, a plane has two degrees of freedom, meaning its position can be described using two independent direction vectors. Common scenarios in IB questions include finding the plane through three points, determining the angle between two planes, and calculating the distance from a point to a plane.
在三维坐标几何中，平面定义为满足一个线性方程的所有点(x, y, z)的集合。从概念上讲，它是二维中直线的三维对应。与具有一个自由度的直线不同，平面有两个自由度，这意味着它的位置可以用两个独立的方向向量来描述。IB考题中常见的情形包括：求过三点的平面、求两平面之间的夹角，以及计算点到平面的距离。
2. Vector and Parametric Equations of a Plane | 平面的向量方程与参数形式
The vector equation of a plane is expressed as r = a + λd₁ + μd₂, where a is the position vector of a known point on the plane, and d₁ and d₂ are two non-parallel direction vectors lying in the plane. The scalars λ (lambda) and μ (mu) are real parameters. This form highlights that any point on the plane can be reached by moving from the reference point a along a linear combination of the two direction vectors. Expanding the vector equation into components yields the parametric equations:
平面的向量方程表示为 r = a + λd₁ + μd₂，其中 a 是平面上一个已知点的位置向量，而 d₁ 和 d₂ 是平面内的两个不共线的方向向量。标量 λ (lambda) 和 μ (mu) 为实数参数。这一形式突出了平面上的任意点都可以从参考点 a 出发，沿着两个方向向量的线性组合到达。将向量方程展开为分量形式，即得参数方程：
x = a₁ + λd₁₁ + μd₂₁, y = a₂ + λd₁₂ + μd₂₂, z = a₃ + λd₁₃ + μd₂₃
Here (a₁, a₂, a₃) are the coordinates of the given point, and (d₁₁, d₁₂, d₁₃) and (d₂₁, d₂₂, d₂₃) are the components of d₁ and d₂. This representation is particularly useful when the plane is defined by three points, as the direction vectors can be taken as the differences between them.
这里 (a₁, a₂, a₃) 是已知点的坐标，(d₁₁, d₁₂, d₁₃) 和 (d₂₁, d₂₂, d₂₃) 分别是 d₁ 和 d₂ 的分量。当平面由三个点确定时，这种表示尤为方便，因为方向向量可以直接取点之间的差向量。
3. Scalar Product (Point-Normal) Form | 点法式（标量积形式）
If a plane is specified by a normal vector n (perpendicular to the plane) and a point A with position vector a, then for any point r on the plane, the vector r - a lies in the plane and is orthogonal to n. Hence (r - a) · n = 0, which simplifies to:
如果一个平面由法向量 n（垂直于平面）和位置向量为 a 的一点A确定，那么对于平面上的任意点 r，向量 r - a 落在平面内并与 n 正交。因此 (r - a) · n = 0，化简得：
r · n = a · n
This is known as the scalar product form or point-normal form. The right-hand side a · n is a constant, often denoted by d. This single equation captures the defining property of the plane: every point on the plane yields the same dot product with the normal vector. It is extremely useful for quickly testing whether a point lies on the plane and for deriving other forms.
这就是标量积形式，也称点法式。等式右边 a · n 是一个常数，通常用 d 表示。这个简洁的方程抓住了平面的定义特性：平面上的每一点与法向量的点积都相同。点法式在快速检验点是否在平面上以及推导其他形式时非常有用。
4. Cartesian Equation of a Plane | 平面的笛卡尔方程
Let the normal vector be n = (a, b, c) and the position vector of a general point be r = (x, y, z). Substituting into r · n = d and expanding gives the Cartesian equation:
设此法向量为 n = (a, b, c)，一般点的位置向量为 r = (x, y, z)。代入 r · n = d 并展开，即得笛卡尔方程：
ax + by + cz = d
Here d = a · n = ax₀ + by₀ + cz₀, where (x₀, y₀, z&# ten the requested final form, but it can be derived from any of the other representations.
这里 d = a · n = ax₀ + by₀ + cz₀，其中 (x₀, y₀, z₀) 是平面上一个已知点的坐标。系数 a, b, c 是法向量的分量；因此笛卡尔形式立刻揭示出平面的一个法向量。在IB考试中，笛卡尔方程通常是要求的最终形式，但它可以从任何其他表示推导得到。
5. Finding the Equation from Three Points | 由三点求平面方程
Given three non-collinear points A, B, C with position vectors a, b, c, we can find the equation of the plane they define. First, form two direction vectors within the plane: AB = b - a and AC = c - a. Then a normal vector is obtained from the cross product:
给定三个不共线的点 A, B, C，其位置向量为 a, b, c，我们可以求出它们所确定的平面方程。首先，构造平面内的两个方向向量：AB = b - a 和 AC = c - a。然后，一个法向量可通过叉乘得到：
n = AB × AC
Once we have n = (n₁, n₂, n₃), we can compute d = n · a (using point A) and write the Cartesian equation as n₁x + n₂y + n₃z = d. For example, if A(1, 2, 3), B(3, 1, 2), C(2, 3, 1), then AB = (2, -1, -1), AC = (1, 1, -2). Their cross product yields n e is x + y + z = 6. This method is standard and emphasizes the power of the cross product.
一旦得到 n = (n₁, n₂, n₃)，就可以令 d = n · a（用点A），从而写出笛卡尔方程 n₁x + n₂y + n₃z = d。例如，若 A(1, 2, 3), B(3, 1, 2), C(2, 3, 1)，则 AB = (2, -1, -1)，AC = (1, 1, -2)。它们的叉乘得 n = (3, 3, 3)，可约简为 (1, 1, 1)。代入点 A 得 d = 1+2+3 = 6，因此平面方程为 x + y + z = 6。此方法标准，并凸显了向量叉乘的威力。
6. Angle Between Two Planes | 两平面之间的夹角
The angle θ between two planes is defined as the acute angle between their normal vectors. If the planes have normals n₁ and n₂, then:
两平面之间的夹角 θ 定义为它们的法向量所夹的锐角。若两平面的法向量为 n₁
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图