Equation and Properties of a Plane in Space | 空间平面的方程与性质

📚 Equation and Properties of a Plane in Space | 空间平面的方程与性质

In three-dimensional coordinate geometry, a plane is a flat, two-dimensional surface that extends infinitely. It is determined by a point and a direction perpendicular to the surface, known as the normal vector. Mastering the equations and properties of planes is a key skill in IB Mathematics, especially for the Vector Geometry topic.

在三维坐标几何中，平面是一个平坦的、无限延伸的二维曲面。它由一点和一个垂直于该面的方向（即法向量）唯一确定。掌握平面的方程与性质是IB数学中的关键技能，尤其在向量几何部分。

1. What is a Plane? | 什么是平面？

A plane is the set of all points P(x, y, z) such that the vector from a fixed point to P is perpendicular to a fixed vector n, called the normal vector.

平面是所有满足如下条件的点 P(x, y, z) 的集合：从某一定点到 P 的向量垂直于一个固定向量 n，这个向量称为法向量。

The normal vector completely determines the orientation of the plane, while the fixed point fixes its position in space.

法向量完全决定了平面的方向，而定点则确定了平面在空间中的位置。

2. Point-Normal Form of a Plane | 平面的点法式方程

Let n = (a, b, c) be a nonzero normal vector and P₀(x₀, y₀, z₀) be a point on the plane. For any point P(x, y, z) on the plane, the vector P₀P = (x − x₀, y − y₀, z − z₀) lies in the plane, so it is perpendicular to n.

设 n = (a, b, c) 为非零法向量，P₀(x₀, y₀, z₀) 为平面上一点。对于平面上任意一点 P(x, y, z)，向量 P₀P = (x − x₀, y − y₀, z − z₀) 位于平面内，因此它垂直于 n。

Using the dot product, we obtain the point-normal equation:

利用点积，我们得到点法式方程：

a(x − x₀) + b(y − y₀) + c(z − z₀) = 0

This equation is satisfied by exactly the points on the plane.

该方程恰好被平面上的所有点满足。

3. General (Cartesian) Equation of a Plane | 平面的一般式（笛卡尔）方程

Expanding the point-normal form gives the general equation of a plane:

将点法式展开即可得到平面的一般式方程：

ax + by + cz + d = 0

where d = −(ax₀ + by₀ + cz₀). Conversely, any equation of this form with (a, b, c) not all zero represents a plane with normal vector n = (a, b, c).

其中 d = −(ax₀ + by₀ + cz₀)。反之，任何这种形式的方程在 (a, b, c) 不全为零时都表示一个平面，其法向量为 n = (a, b, c)。

The value of d affects the position of the plane: if d = 0, the plane passes through the origin.

常数 d 影响平面的位置：若 d = 0，则平面过原点。

4. Parametric Form of a Plane | 平面的参数方程

A plane can also be described using two non-parallel direction vectors u and v lying in the plane. If A is a point on the plane, then any point P on the plane can be written as:

平面也可以用两个位于平面内且不平行的方向向量 u 和 v 来描述。若 A 是平面上一点，则平面上任一点 P 可写成：

r = a + λu + μv, λ, μ ∈ ℝ

where r = (x, y, z) and a is the position vector of A.

其中 r = (x, y, z)，a 是点 A 的位置向量。

The normal vector can be obtained by the cross product n = u × v.

法向量可以通过叉积 n = u × v 得到。

5. Special Planes | 特殊位置的平面

Certain equations correspond to planes parallel to axes or coordinate planes.

某些方程对应平行于坐标轴或坐标平面的平面。

Planes parallel to the yz-plane: x = k (normal vector (1, 0, 0))

平行于 yz 平面的平面：x = k（法向量 (1, 0, 0)）
Planes parallel to the xz-plane: y = k (normal vector (0, 1, 0))

平行于 xz 平面的平面：y = k（法向量 (0, 1, 0)）
Planes parallel to the xy-plane: z = k (normal vector (0, 0, 1))

平行于 xy 平面的平面：z = k（法向量 (0, 0, 1)）
Planes parallel to the z-axis: ax + by + d = 0 (normal vector (a, b, 0))

平行于 z 轴的平面：ax + by + d = 0（法向量 (a, b, 0)）

Recognising these cases helps in sketching and solving geometric problems quickly.

识别这些特殊情况有助于快速作图并解决几何问题。

6. Angle Between Two Planes | 两平面的夹角

The angle θ between two planes is the acute angle between their normal vectors. If the planes have normals n₁ and n₂, then:

两个平面的夹角 θ 是它们法向量之间的锐角。若两平面的法向量为 n₁ 和 n₂，则：

cos θ = |n₁ · n₂| / (|n₁| |n₂|)

The absolute value ensures that we take the acute angle. Two planes are perpendicular if n₁ · n₂ = 0, and parallel if n₁ × n₂ = 0.

取绝对值保证了我们得到锐角。两个平面垂直的条件是 n₁ · n₂ = 0；平行的条件是 n₁ × n₂ = 0。

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

Given a plane ax + by + cz + d = 0 and a point P₀(x₀, y₀, z₀), the perpendicular distance D is:

给定平面 ax + by + cz + d = 0 和点 P₀(x₀, y₀, z₀)，点到平面的垂直距离 D 为：

D = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)

This formula is derived by projecting the vector from any known point on the plane to P₀ onto the normal vector n.

该公式通过将平面上任意已知点到 P₀ 的向量投影到法向量 n 上推导得出。

Notice that distance is always non-negative due to the absolute value.

注意由于取绝对值，距离总是非负的。

8. Relative Positions of Two Planes | 两平面的位置关系

Two planes can intersect along a line, be parallel, or be coincident.

两个平面可能相交于一条直线、平行或重合。

If their normal vectors are scalar multiples, the planes are parallel or coincident. If additionally the constant terms have the same ratio, they are coincident.

若它们的法向量成比例，则两平面平行或重合。若常数项也满足相同比例，则两平面重合。
If the normal vectors are not parallel, the planes intersect in a straight line.

若法向量不平行，则两平面相交于一条直线。

For example, x + y + z = 1 and 2x + 2y + 2z = 2 represent the same plane, while x + y + z = 1 and x + y + z = 2 are parallel.

例如，x + y + z = 1 与 2x + 2y + 2z = 2 表示同一个平面，而 x + y + z = 1 与 x + y + z = 2 是平行的。

9. Intersection of a Line and a Plane | 直线与平面的位置关系

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Equation and Properties of a Plane in Space | 空间平面的方程与性质