IB Mathematics: Relative Positions of Two Planes | IB数学：平面与平面的位置关系

📚 IB Mathematics: Relative Positions of Two Planes | IB数学：平面与平面的位置关系

In three-dimensional space, two planes can be arranged in a limited number of ways. Understanding these arrangements is essential for solving problems in vector geometry, especially in the IB Mathematics Analysis & Approaches and Applications & Interpretation courses.

在三维空间中，两个平面的位置关系只有有限几种。理解这些关系是解决向量几何问题的基础，尤其对于IB数学分析与方法（AA）以及应用与解释（AI）课程都至关重要。

1. Plane Equations | 平面方程回顾

Before discussing the relative positions of two planes, we must first be comfortable with the equation of a plane. The general Cartesian equation of a plane is \(ax+by+cz=d\), which can also be written as \(\mathbf{r}\cdot\mathbf{n}=d\), where \(\mathbf{n}=(a,b,c)\) is a normal vector perpendicular to the plane.

在讨论两个平面的位置关系之前，我们必须熟悉平面的方程。平面的一般笛卡尔方程为 \(ax+by+cz=d\)，也可以写成向量形式 \(\mathbf{r}\cdot\mathbf{n}=d\)，其中 \(\mathbf{n}=(a,b,c)\) 是该平面的法向量，垂直于平面。

Another common form is the parametric form, where a point on the plane and two non-parallel direction vectors are used. However, the normal vector form is most convenient for analysing relationships between planes.

另一种常见形式是参数方程，它使用平面上一点和两个不平行方向向量。然而，法向量形式最适合用来分析平面之间的关系。

2. Two Possible Relative Positions | 两种基本位置关系

In three-dimensional space, two distinct planes can either be parallel or intersect along a straight line. There is no third possibility: if they are not parallel, they must intersect, and their intersection is always a line.

在三维空间中，两个不同的平面要么平行，要么相交于一条直线。不存在第三种可能：如果不平行，则必然相交，且交线是一条直线。

If the planes are parallel and also coincide, they are considered the same plane, which is a special case of parallelism. In the IB syllabus, we usually distinguish between ‘parallel and distinct’ and ‘coincident’ planes.

如果两个平面平行且完全重合，则它们是同一个平面，这是平行的一种特殊情况。在IB课程中，我们通常区分“平行且不同”与“重合”两种情形。

Parallel planes – no intersection.
Intersecting planes – intersection is a line.

平行平面——没有交点。
相交平面——交线为一条直线。

3. Parallel Planes | 平行平面的判定

Two planes are parallel if and only if their normal vectors are parallel (scalar multiples of each other). That is, given planes \(P_1: a_1x+b_1y+c_1z=d_1\) and \(P_2: a_2x+b_2y+c_2z=d_2\), the planes are parallel if there exists a constant \(k\) such that \((a_1,b_1,c_1)=k(a_2,b_2,c_2)\).

两个平面平行的充要条件是它们的法向量平行（互为标量倍数）。即给定平面 \(P_1: a_1x+b_1y+c_1z=d_1\) 和 \(P_2: a_2x+b_2y+c_2z=d_2\)，若存在常数 \(k\)，使得 \((a_1,b_1,c_1)=k(a_2,b_2,c_2)\)，则两平面平行。

After confirming parallel normals, compare the constant terms. If the ratios are equal including \(d\), the planes are identical. If not, they are distinct parallel planes.

确认法向量平行后，再比较常数项。如果包含 \(d\) 在内的比值都相等，则两平面重合；否则它们是两个不同的平行平面。

Planes \(P_1\) and \(P_2\) are parallel ⇔ \(\mathbf{n}_1 \times \mathbf{n}_2 = \mathbf{0}\)

平面 \(P_1\) 与 \(P_2\) 平行 ⇔ \(\mathbf{n}_1 \times \mathbf{n}_2 = \mathbf{0}\)

For example, \(x+2y-3z=4\) and \(2x+4y-6z=10\) are parallel because their normals \((1,2,-3)\) and \((2,4,-6)\) are proportional. Since \(d_1=4\) and \(d_2=10\) do not follow the same ratio (\(2\times4 \neq 10\)), the planes are distinct.

例如，\(x+2y-3z=4\) 与 \(2x+4y-6z=10\) 是平行的，因为它们的法向量 \((1,2,-3)\) 和 \((2,4,-6)\) 成比例。由于 \(d_1=4\) 与 \(d_2=10\) 不满足相同比例（\(2\times4 \neq 10\)），所以这两个平面是互不相同的平面。

4. Distance Between Parallel Planes | 平行平面间的距离

The distance between two parallel planes can be found by choosing a point on one plane and calculating its perpendicular distance to the other plane. If the planes are \(ax+by+cz=d_1\) and \(ax+by+cz=d_2\), then the distance is given by the formula below.

两个平行平面之间的距离可以通过在一个平面上取一点，然后计算该点到另一个平面的垂直距离来求得。若两个平面分别为 \(ax+by+cz=d_1\) 和 \(ax+by+cz=d_2\)，则距离由以下公式给出。

Distance = \(\frac{|d_1 – d_2|}{\sqrt{a^2 + b^2 + c^2}}\)

距离 = \(\frac{|d_1 – d_2|}{\sqrt{a^2 + b^2 + c^2}}\)

This formula is not directly in the IB formula booklet but is easily derived from the point-to-plane distance formula. It is often tested in paper 2 questions involving 3D geometry.

该公式并不直接在IB公式手册中，但可以很容易地从点到平面的距离公式推导出来。在试卷二的3D几何题中经常考查。

5. Intersecting Planes | 相交平面

If two planes are not parallel, their intersection is a straight line. To find the line of intersection, we solve the two plane equations simultaneously. The solution set will be expressed in terms of a parameter, which is exactly the parametric equation of the line.

如果两个平面不平行，它们的交线是一条直线。要求交线，我们需要联立两个平面方程。解集可以用一个参数表示，而这正是直线的参数方程。

For example, consider the planes \(x+y+z=1\) and \(x-y+z=2\). Subtracting the equations gives \(2y=-1\), so \(y=-\frac12\). Substituting back gives \(x+z=\frac32\), hence the line can be written as \(x=t\), \(z=\frac32-t\), \(y=-\frac12\).

例如，考虑平面 \(x+y+z=1\) 和 \(x-y+z=2\)。两式相减得 \(2y=-1\)，所以 \(y=-\frac12\)。代回得 \(x+z=\frac32\)，因此直线可写成 \(x=t\)，\(z=\frac32-t\)，\(y=-\frac12\)。

Line of intersection: \(\mathbf{r} = \mathbf{p} + t\mathbf{d}\)

交线：\(\mathbf{r} = \mathbf{p} + t\mathbf{d}\)

Here \(\mathbf{p}\) is a particular point on the line and \(\mathbf{d}\) is the direction vector. The direction vector of the intersection line must be perpendicular to both plane normals, so it can be found using the cross product \(\mathbf{n}_1 \times \mathbf{n}_2\).

其中 \(\mathbf{p}\) 是直线上某一点，\(\mathbf{d}\) 是方向向量。交线的方向向量必须同时垂直于两个平面的法向量，因此可以通过叉积 \(\mathbf{n}_1 \times \mathbf{n}_2\) 求得。

6. Direction Vector of the Intersection Line | 交线的方向向量

Given two planes with normals \(\mathbf{n}_1\) and \(\mathbf{n}_2\), the direction vector \(\mathbf{d}\) of the line of intersection is \(\mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2\). This is because the line lies in both planes, so its direction must be orthogonal to both normals.

对于法向量分别为 \(\mathbf{n}_1\) 和 \(\mathbf{n}_2\) 的两个平面，交线的方向向量为 \(\mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2\)。因为交线同时位于两个平面内，所以其方向必须与两个法向量都正交。

Let us compute an example. Take \(P_1: x+y+z=1\) and \(P_2: x-y+z=2\). Then \(\mathbf{n}_1=(1,1,1)\) and \(\mathbf{n}_2=(1,-1,1)\). Their cross product is:

我们来看一个例子。设 \(P_1: x+y+z=1\)，\(P_2: x-y+z=2\)。则 \(\mathbf{n}_1=(1,1,1)\)，\(\mathbf{n}_2=(1,-1,1)\)。它们的叉积为：

\(\mathbf{n}_1 \times \mathbf{n}_2 = (1,1,1)\times(1,-1,1) = (2,0,-2)\)

Thus the line has direction vector \((2,0,-2)\), which can be simplified to \((1,0,-1)\). Indeed, from the earlier parameterisation we found \(x=t, y=-\frac12, z=\frac32-t\), whose direction is \((1,0,-1)\).

因此交线的方向向量为 \((2,0,-2)\)，可以简化为 \((1,0,-1)\)。事实上，前面我们得到的参数化 \(x=t, y=-\frac12, z=\frac32-t\) 的方向就是 \((1,0,-1)\)。

7. Angle Between Two Planes | 两平面夹角（二面角）

The angle between two planes is defined as the acute angle between their normals. If \(\theta\) is the angle between the normals \(\mathbf{n}_1\) and \(\mathbf{n}_2\), then the angle between the planes is either \(\theta\) or \(180^\circ – \theta\), whichever is acute. In practice, we take the acute angle using the absolute value of the dot product.

两个平面的夹角定义为它们的法向量之间的锐角。若 \(\theta\) 是法向量 \(\mathbf{n}_1\) 与 \(\mathbf{n}_2\) 之间的夹角，则两平面的夹角为 \(\theta\) 或 \(180^\circ – \theta\) 中较小的那个。实际计算中，我们通过点积的绝对值来取锐角。

\(\cos \theta = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{|\mathbf{n}_1||\mathbf{n}_2|}\)

The absolute value ensures that the angle is between \(0^\circ\) and \(90^\circ\). If the dot product is zero, the planes are perpendicular.

加上绝对值可以确保夹角在 \(0^\circ\) 到 \(90^\circ\) 之间。如果点积为零，则两平面互相垂直。

8. Perpendicular Planes | 垂直平面的判定

Two planes are perpendicular if their normal vectors are perpendicular, i.e. \(\mathbf{n}_1 \cdot \mathbf{n}_2 = 0\). This is a direct consequence of the angle formula above.

两个平面垂直当且仅当它们的法向量垂直，即 \(\mathbf{n}_1 \cdot \mathbf{n}_2 = 0\)。这是上述夹角公式的直接推论。

For instance, \(2x-y+z=5\) and \(x+2y=3\) are perpendicular because \((2,-1,1)\cdot(1,2,0)=2-2+0=0\).

例如，\(2x-y+z=5\) 与 \(x+2y=3\) 垂直，因为 \((2,-1,1)\cdot(1,2,0)=2-2+0=0\)。

Perpendicular planes appear frequently in problems involving cubes, rectangular boxes, and other 3D solids. Remember that a plane is perpendicular to another plane if it contains a line perpendicular to that plane.

垂直平面在涉及立方体、长方体和其他三维立体的问题中经常出现。记住：如果一个平面包含另一个平面的垂线，那么这两个平面垂直。

9. Solving Problems with Planes | 利用平面关系解题

In IB exams, you may be asked to determine whether two planes are parallel, perpendicular, or neither, and to find the angle between them. For two planes given in Cartesian form, compare their normals first. If the normals are parallel, check the constants to see if they are coincident. If the normals are not parallel, the planes intersect.

在IB考试中，你可能会被要求判断两个平面是平行、垂直还是其他相交关系，并求它们之间的夹角。对于给定笛卡尔方程的两个平面，首先比较它们的法向量。如果法向量平行，则检查常数项判断是否重合；如果法向量不平行，则两平面相交。

When asked to find the line of intersection, use the cross product for the direction vector and then find one point on the line by setting one variable equal to zero and solving the remaining system.

当要求交线时，用叉积求方向向量，然后通过令一个变量等于零并求解剩余方程组，来找到直线上的一点。

Relationship	Condition on normals
Parallel and distinct	\(\mathbf{n}_1 = k\mathbf{n}_2\), but \(d_1 \neq k d_2\)
Coincident	\(\mathbf{n}_1 = k\mathbf{n}_2\), and \(d_1 = k d_2\)
Intersecting (including perpendicular)	\(\mathbf{n}_1\) and \(\mathbf{n}_2\) not parallel

位置关系	法向量条件
平行且不同	\(\mathbf{n}_1 = k\mathbf{n}_2\)，但 \(d_1 \neq k d_2\)
重合	\(\mathbf{n}_1 = k\mathbf{n}_2\)，且 \(d_1 = k d_2\)
相交（包括垂直）	\(\mathbf{n}_1\) 与 \(\mathbf{n}_2\) 不平行

10. Worked Example | 典型例题

Consider the planes \(P_1: x-2y+2z=3\) and \(P_2: 2x-4y+4z=7\). Are they parallel? If so, find the distance between them.

考虑平面 \(P_1: x-2y+2z=3\) 和 \(P_2: 2x-4y+4z=7\)。它们是否平行？若平行，求它们之间的距离。

First, compare normals: \((1,-2,2)\) and \((2,-4,4)=2(1,-2,2)\), so they are parallel. Since \(2 \times 3 = 6 \neq 7\), they are distinct. To use the distance formula, we rewrite \(P_2\) as \(x-2y+2z=\frac72\). Then the distance is:

首先比较法向量：\((1,-2,2)\) 和 \((2,-4,4)=2(1,-2,2)\)，因此它们平行。由于 \(2 \times 3 = 6 \neq 7\)，所以它们是两个不同平面。为了使用距离公式，将 \(P_2\) 改写为 \(x-2y+2z=\frac72\)。则距离为：

\(\frac{|3 – \frac72|}{\sqrt{1^2+(-2)^2+2^2}} = \frac{\frac12}{3} = \frac16\)

Thus the distance between the two parallel planes is \(\frac16\) units.

因此这两个平行平面之间的距离为 \(\frac16\) 个单位。

11. Common Mistakes | 常见错误提醒

Forgetting that parallel normals do not necessarily mean distinct planes; always check the constants.
Using the angle between normals directly without taking absolute value, which may give the obtuse angle.
Confusing the direction vector of the intersection line with the normals themselves.

忘记法向量平行并不一定意味着平面不同；务必检查常数项。
直接使用法向量夹角而不取绝对值，可能得到钝角。
将交线的方向向量与法向量本身混淆。

To avoid these errors, systematically write down the normals, compare them, and always draw a quick sketch of the situation in 3D if possible.

为避免这些错误，应系统性地写出法向量、进行比较，并在可能的情况下在三维空间中快速画出示意图。

12. Summary | 总结

Two planes in space can be parallel (possibly coincident) or intersecting along a line. Parallelism is determined by proportional normal vectors; perpendicularity is determined by a zero dot product; the angle between planes is the acute angle between normals; and the intersection line direction is given by the cross product of normals.

空间中的两个平面要么平行（可能重合），要么相交于一条直线。平行由法向量成比例确定；垂直由点积为零确定；两平面夹角是法向量之间的锐角；交线方向由法向量的叉积给出。

Mastering these concepts will help you solve vector geometry problems efficiently in both the analysis and applications courses. Practice with past paper questions to become confident in identifying relationships between planes.

掌握这些概念将帮助你在分析与应用课程中高效解决向量几何问题。通过练习历年真题，你将能够自信地判断平面之间的关系。

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IB Mathematics: Relative Positions of Two Planes | IB数学：平面与平面的位置关系