Plane and Plane: Intersections, Angles and Distances | 平面与平面：相交、夹角与距离

📚 Plane and Plane: Intersections, Angles and Distances | 平面与平面：相交、夹角与距离

In three-dimensional geometry, understanding how two planes relate to each other is a cornerstone of vector analysis. This article explores the possible configurations of a pair of planes, including parallel, coincident, and intersecting cases. We derive methods for finding the line of intersection, calculating the angle between planes, measuring the distance between parallel planes, and solving IB-style problems using normal vectors and algebraic techniques.

在三维几何中，理解两个平面之间的相互关系是向量分析的基础。本文将探讨两个平面之间可能的位置关系，包括平行、重合和相交的情形。我们将推导求相交直线的方法、计算两个平面之间的夹角、度量两个平行平面之间的距离，并运用法向量和代数方法解决IB风格的题目。

1. Equation of a Plane: A Quick Review | 平面方程的快速回顾

A plane in 3D space can be expressed in vector form as r · n = a · n, where n is a vector normal to the plane and a ivalently, in Cartesian form it is ax + by + cz = d, where the normal vector is (a, b, c).

三维空间中的平面可以用向量形式表示为 r · n = a · n，其中 n 是平面的一条法向量，a 是平面上一个已知点的位置向量。等价的笛卡尔形式为 ax + by + cz = d，法向量就是 (a, b, c)。

Any scalar multiple of the normal vector is also normal to the plane, and the constant term d determines the plane's position relative to the origin. Changing d shifts the plane parallel to itself, while altering the normal direction tilts the plane.

法向量的任何非零标量倍数也是该平面的法向量，常数项 d 决定了平面相对于原点的位置。改变 d 会使平面平行移动，而改变法向量方向则会使平面倾斜。

2. Normal Vectors and Plane Orientation | 法向量与平面的方向

A plane is completely oriented by its normal vector. Two planes are parallel if and only if their normal vectors are parallel (i.e., one is a scalar multiple of the other). If the normal vectors are not parallel, the planes must intersect in a straight line.

一个平面的方向完全由其法向量决定。两个平面平行的充要条件是它们的法向量平行（即，一个法向量是另一个的标量倍数）。如果法向量不平行，则这两个平面必定相交于一条直线。

Moreover, if two planes have parallel normal vectors and share at least one common point, they are coincident — that is, the same plane written in two forms. Distinguishing between parallel and coincident requires checking whether a point from one plane satisfies the equation of the other.

此外，若两个平面的法向量平行且它们至少共享一个公共点，则两个平面重合——即同一个平面的两种写法。区分平行与重合需要检验一个平面上的点是否满足另一个平面的方程。

3. Possible Configurations of Two Planes | 两个平面可能的位置关系

Given two planes Π₁ and Π₂, exactly one of the following holds:

给定两个平面 Π₁ 和 Π₂，必满足以下情况之一：

Case 1: The normal vectors are parallel. If additionally a point of Π₁ lies on Π₂, the planes are coincident. Otherwise they are parallel and distinct, with a constant separation distance.

情况 1：法向量平行。若再加上 Π₁ 上的一点在 Π₂ 上，则两平面重合；否则它们是平行但不重合的，且具有恒定的间距。

Case 2: The normal vectors are not parallel. The planes intersect along a unique straight line. No other possibility exists in three dimensions.

情况 2：法向量不平行。这两个平面沿一条唯一的直线相交。在三维空间中不存在其他可能性。

4. Determining Parallel or Coincident Planes | 判定平行或重合

Take Π₁: a₁x + b₁y + c₁z = d₁ and Π₂: a₂x + b₂y + c₂z = d₂. The normal vectors are n₁ = (a₁, b₁, c₁) and n₂ = (a₂, b₂, c₂). If n₁ = kn₂ for some nonzero scalar k, the planes are parallel or coincident.

设有 Π₁: a₁x + b₁y + c₁z = d₁ 和 Π₂: a₂x + b₂y + c₂z = d₂，法向量分别为 n₁ = (a₁, b₁, c₁) 和 n₂ = (a₂, b₂, c₂)。若存在非零标量 k 使 n₁ = kn₂，则两平面平行或重合。

k d₂ (using the same k), they are coincident; otherwise they are distinct parallel planes. A robust test is to pick a point on Π₁, say by setting two variables to zero and solving for the third, and substitute it into Π₂'s equation — if it satisfies, the planes coincide.

区分平行与重合只需比较常数项：如果 d₁ = k d₂（使用相同的 k），则两平面重合；否则它们是不同的平行平面。一个可靠的检验方法是选取 Π₁ 上的一点（例如令两个变量为零解出第三个），代入 Π₂ 的方程——如果满足，则两平面重合。

5. Line of Intersection When Planes Intersect | 两平面相交时的交线

When normal vectors are not parallel, the two planes intersect in a line. A direction vector d of this intersection line must be perpendicular to both normals, so it can be found by the cross product: d = n₁ × n₂.

当法向量不平行时，两平面相交于一条直线。该交线的方向向量 d 必须同时垂直于两个法向量，因此可通过叉积求得：d = n₁ × n₂。

To fully determine the line, we also need a specific point common to both planes. One method is to set one of the variables x, y, or z to a convenient value (often 0) and solve the remaining two equations simultaneously for the other two variables. This gives a point on both planes.

为完全确定该直线，还需要一个同时属于两个平面的特定点。一种方法是将 x、y 或 z 中的一个设为方便的值（通常取 0），然后联立剩下的两个方程求解另外两个变量，由此得到两平面的一个交点。

6. Step-by-Step Procedure for the Line of Intersection | 求交线的分步方法

Step 1: Write both plane equations in standard form. Compute the normal vectors n₁ and n₂. Verify they are not multiples — otherwise no unique intersection line exists.

步骤 1：将两个平面方程都写成标准形式。计算法向量 n₁ 和 n₂。验证它们不成比例——否则不存在唯一的交线。

Step 2: Compute the direction vector d = n₁ × n₂. Any nonzero cross product works; simplify if needed.

步骤 2：计算方向向量 d = n₁ × n₂。任取叉积的一个非零结果即可；必要时化简。

Step 3: Find a common point by setting z = 0 (if d nient value). Solve the resulting 2×2 system for x and y. Let this point be P(x₀, y₀, z₀).

广告 / Sponsored

步骤 3：通过设 z = 0 求一个公共点（若 d 的 z 分量不为零，通常可行；否则设另一变量为合适的值）。求解得到的 2×2 线性方程组得出 x 和 y。记该点为 P(x₀, y₀, z₀)。

Step 4: Write the vector equation of the line as r = p + λ d or the parametric equations: x = x₀ + λd₁, y = y₀ + λd₂, z = z₀ + λd₃.

步骤 4：写出直线的向量方程 r = p + λ d 或参数方程：x = x₀ + λd₁，y = y₀ + λd₂，z = z₀ + λd₃。

An alternative method solves the system by expressing two variables in terms of a parameter directly from the equations, which yields the same line.

另一种方法是直接从方程组中用一个参数表示两个变量，最终得到同一直线。

7. Angle Between Two Planes | 两个平面之间的夹角

The angle between two intersecting planes is defined as the acute angle between their normal vectors. If n₁ and n₂ are the normals, the angle θ satisfies:

两个相交平面之间的夹角定义为其法向量之间的锐角。若 n₁ 和 n₂ 为法向量，则夹角 θ 满足：

cos θ = |n₁ · n₂| / (|n₁| |n₂|)

parallel (θ = 0°). In all cases, this formula yields the correct angle between the planes.

分子加绝对值保证取锐角（0 ≤ θ ≤ 90°）。若法向量平行，则平面平行（θ = 0°）。所有情况下，该公式均能给出正确的平面夹角。

8. Distance Between Two Parallel Planes | 两个平行平面之间的距离

: ax + by + cz = d₂ (with the same a, b, c), the perpendicular distance D between them is:

给定两个平行平面 Π₁: ax + by + cz = d₁ 和 Π₂: ax + by + cz = d₂（a, b, c 相同），它们之间的垂直距离 D 为：

D = |d₂ − d₁| / √a² + b² + c²

This formula follows directly from the point-to-plane distance expression because picking any point on Π₁ and calculating its distance to Π₂ gives the same result regardless of the point chosen.

该公式直接从点到平面的距离表达式推出，因为从 Π₁ 上任选一点计算其到 Π₂ 的距离，结果均相同。

If the equations are given in a form where the normal vectors are multiples but not identical, first scale one equation so that both normals match. For instance, multiply Π₂ by an appropriate factor so the coefficients of x, y, z match those of Π₁, then apply the formula.

若给出方程的法向量成比例但不相同，需先缩放其中一个方程使两个法向量系数一致。例如将 Π₂ 乘以适当因子使 x, y, z 系数与 Π₁ 相同，再代入公式。

9. Worked Example: Intersection and Angle | 完整例题：求交线与夹角

3 and Π₂: x + y − 2z = 1. Find the line of intersection and the angle between the planes.

给定平面 Π₁：2x − y + z = 3 和 Π₂：x + y − 2z = 1。求交线以及两平面之间的夹角。

Normals: n₁ = (2, −1, 1), n₂ = (1, 1, −2). They are not multiples, so planes intersect.

法向量：n₁ = (2, −1, 1)，n₂ = (1, 1, −2)。不成比例，因此两平面相交。

Direction vector: d = n₁ × n (1)(1), (1)(1) − (2)(−2), (2)(1) − (−1)(1) ) = (2 − 1, 1 − (−4), 2 − (−1)) = (1, 5, 3). So d = (1, 5, 3).

方向向量：d = n₁ × n₂ = (1, 5, 3)。

Common point: y = 1. Adding gives 3x = 4 → x = 4/3, then y = 1 − x = −1/3. So point P = (4/3, −1/3, 0).

公共点：设 z = 0，方程组变为 2x − y = 3 和 x + y = 1。相加得 3x = 4 → x = 4/3，则 y = 1 − x = −1/3。因此点 P = (4/3, −1/3, 0)。

Line of intersection: r = (4/3, −1/3, 0) + λ (1, 5, 3).

交线：r = (4/3, −1/3, 0) + λ (1, 5, 3)。

Angle: |n₁| = √4+1+1 = √6, |n₂| = √1+1+4 = √6. Dot product = 2×1 + (−1)×1 + 1×(−2) = 2 − 1 − 2 = −1. cos θ = |−1|/(√6 × √6) = 1/6. Hence θ = arccos(1/6) ≈ 80.4°.

夹角：|n₁| = √6，|n₂| = √6，点积 = −1，cos θ = 1/6，θ = arccos(1/6) ≈ 80.4°。

10. Common Pitfalls and Exam Tips | 常见误区与考试技巧

Pitfall 1: Forgetting the absolute value in the angle formula. Always use |n₁ · n₂| to obtain the acute angle; otherwise you might report the obtuse supplement.

误区 1：夹角公式中忘记加绝对值。务必使用 |n₁ · n₂| 求锐角，否则可能会错误地给出钝角补角。

Pitfall 2: Confusing parallel with coincident planes. Always test a point to confirm whether planes coincide when their normals are multiples.

误区 2：混淆平行与重合。当法向量成比例时，务必检验一个点来确认两平面是否重合。

Pitfall 3: In distance calculations, using normals that are multiples but not identical. Scale the equations first so that the coefficients of x, y, z match exactly before applying the distance formula.

误区 3：在距离计算中用了成比例但不相同的法向量。应先对方程进行缩放使 x, y, z 的系数完全一致，再应用距离公式。

Exam tip: When finding the intersection line, always double-check your common point satisfies both original plane equations. A quick mental substitution can prevent sign errors.

考试技巧：求交线时，务必重新核验公共点是否满足两个原平面方程。快速心算代入可以防止符号错误。

or y = 0 instead of z = 0 if the direction vector has a zero component in that coordinate — this avoids an inconsistent system.

技巧：如果方向向量在某个坐标上的分量为零，有时巧妙地设 x = 0 或 y = 0 而非 z = 0 可以避免产生矛盾方程组。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）