📚 Constructing Line Equations in 3D Space | 三维空间直线方程的构建方法
In three-dimensional space, a straight line is uniquely determined by a fixed point and a direction vector. Unlike lines in 2D, a 3D line cannot be described by a single linear equation; instead, it requires a vector, parametric, or symmetric equation system. This article explores the core methods for constructing these equations, tailored to IB Mathematics Analysis and Approaches HL.
在三维空间中，一条直线由一定点和一方向向量唯一确定。与二维直线不同，三维直线不能由单个线性方程描述，而需要使用向量方程、参数方程或对称方程系统。本文将围绕IB数学分析与方法HL的考点，系统讲解这些方程的构建方法。
1. Core Idea: Point + Direction Vector | 核心思想：点 + 方向向量
To build any line equation in 3D, you need two ingredients: a known point A on the line (with position vector a) and a direction vector d parallel to the line. Every point P on the line can then be reached by starting at A and moving a scalar multiple of d.
要构建三维空间中的任意直线方程，需要两个要素：直线上已知一点 A（位置向量为 a）以及平行于直线的方向向量 d。直线上任意一点 P 都可以从 A 出发，沿 d 方向移动某个标量倍数而得到。
P = A + t·d or r = a + t·d (t ∈ ℝ)
Here t is a parameter. Different values of t correspond to different points on the line. This underlying relation is the basis of all three standard equation forms.
其中 t 是参数。不同的 t 值对应直线上不同的点。这一基本关系是三种标准方程形式的基础。
2. Vector Equation of a Line | 直线的向量方程
The vector equation directly states the position vector r of a general point on the line:
向量方程直接给出直线上一般点的位置向量 r：
r = a + t·d
For example, if a line passes through A(1, -2, 3) and has direction vector d = (4, 1, -2), its vector equation is
例如，若一条直线经过点 A(1, -2, 3)，方向向量为 d = (4, 1, -2)，则其向量方程为
r = (1, -2, 3) + t(4, 1, -2)
Here r = (x, y, z). In IB, you may write this as (x, y, z)ᵀ = (1, -2, 3)ᵀ + t (4, 1, -2)ᵀ.
这里 r = (x, y, z)。在IB中，你可以写成 (x, y, z)ᵀ = (1, -2, 3)ᵀ + t (4, 1, -2)ᵀ。
3. Parametric Equations | 参数方程
By equating components in the vector equation, we obtain three separate equations for x, y and z in terms of the parameter t:
将向量方程中的各分量分别相等，即可得到关于参数 t 的三个独立方程：
x = x₀ + t·d₁, y = y₀ + t·d₂, z = z₀ + t·d₃
Using the previous example, the parametric equations are:
使用前面的例子，参数方程为：
x = 1 + 4t, y = -2 + t, z = 3 - 2t
Parametric form is extremely useful because it gives explicit coordinates for every point on the line. It also simplifies calculations of intersections and distances.
参数形式非常实用，因为它给出了直线上每个点的显式坐标，同时简化了交点与距离的计算。
4. Symmetric (Cartesian) Equations | 对称（笛卡尔）方程
If each component of the direction vector is non-zero, we can eliminate the parameter t by solving each parametric equation for t and equating the results:
若方向向量的每个分量均不为零，我们可以解出每个参数方程中的 t，再令它们相等，从而消去参数 t：
x - x₀d₁ = y - y₀d₂ = z - z₀d₃
For the example above, the symmetric equations are
对于上述例子，对称方程为
x - 14 = y + 21 = z - 3-2
Note that if a directional component is zero, the corresponding numerator must equal zero as well. For instance, if d₂ = 0, then y = y₀ and the symmetric form becomes a pair of equalities plus a separate condition.
注意若方向向量的某分量为零，则对应分子也必须等于零。例如，若 d₂ = 0，则 y = y₀，对称形式变为一组等量关系外加一个独立条件。
5. Building a Line from Two Points | 由两点构建直线
Given two distinct points A and B, the direction vector can be found by subtracting their position vectors: d = b - a. Then choose either point to write the equation.
已知两个不同点 A 和 B，方向向量可通过它们的位置向量相减得到：d = b - a。然后选择其中任意一点写出方程。
Let A(2, -1, 5) and B(0, 3, -1). Then
设 A(2, -1, 5)，B(0, 3, -1)。则
d = b - a = (0-2, 3-(-1), -1-5) = (-2, 4, -6)
Using point A, the vector equation is r = (2, -1, 5) + t(-2, 4, -6). Symmetric form:
以点 A 为基础，向量方程为 r = (2, -1, 5) + t(-2, 4, -6)。对称形式为：
x - 2-2 = y + 14 = z - 5-6
Always check that both points satisfy the final equation. This verifies that the direction vector is correct.
务必验证两个点是否都满足最终方程，以确认方向向量无误。
6. Direction Ratios and Direction Cosines | 方向比与方向余弦
Direction ratios are the components of any vector parallel to the line. They are not unique; multiplying by a non-zero scalar gives another valid set. For example, d = (2, -4, 6) and d = (1, -2, 3) represent the same direction.
方向比是平行于直线的任意向量的分量。它们不唯一；乘以非零标量仍得到有效的一组方向比。例如，d = (2, -4, 6) 与 d = (1, -2, 3) 表示同一直线方向。
Direction cosines are the cosines of the angles between the line and the positive x, y, and z axes. They are obtained by normalising the direction vector:
方向余弦是直线与 x、y、z 轴正方向夹角之余弦，由方向向量单位化得到：
cos α = d₁⁄|d|, cos β = d₂⁄|d|, cos γ = d₃⁄|d|
These satisfy cos²α + cos²β + cos²γ = 1. In IB, direction cosines sometimes appear in vector geometry problems, so be ready to compute them from a given direction vector.
这些方向余弦满足 cos²α + cos²β + cos²γ = 1。在IB考试中，方向余弦偶见于向量几何题中，因此应熟练掌握从给定方向向量计算它们。
7. Converting Between Different Equation Forms | 不同方程形式之间的转换
The three forms (vector, parametric, symmetric) are interchangeable. To convert from symmetric to vector form, set the common ratio equal to t, then express x, y, z as linear expressions in t.
三种形式（向量、参数、对称）可以相互转换。从对称形式到向量形式，令公共比值等于 t，然后将 x、y、z 表示为 t 的线性表达式。
For example, given x - 12 = y-3 = z + 41, set this equal to t:
例如，已知 x - 12 = y-3 = z + 41，令它等于 t：
x = 1 + 2t, y = -3t, z = -4 + t
This gives the vector equation r = (1, 0, -4) + t(2, -3, 1). Notice that the fixed point corresponds to t = 0.
由此得到向量方程 r = (1, 0, -4) + t(2, -3, 1)。注意 t = 0 时对应定点。
8. Parallel and Coincident Lines | 平行直线与重合直线
Two lines are parallel if their direction vectors are scalar multiples of each other. If they are parallel and share a common point, they are coincident (the same line).
若两直线的方向向量成比例，则它们平行。若平行且共有一个公共点，则它们重合（即同一条直线）。
Given L₁: r = (1, 2, 3) + s(2, -1, 4) and L₂: r = (3, 1, 7) + t(-4, 2, -8), we see d₂ = -2d₁, so the lines are parallel. Since (3,1,7) lies on L₁? Check: set s=1 gives (3,1,7), yes, hence they are actually coincident.
已知 L₁: r = (1, 2, 3) + s(2, -1, 4) 和 L₂: r = (3, 1, 7) + t(-4, 2, -8)，观察得 d₂ = -2d₁，所以两直线平行。又 (3,1,7) 是否在 L₁ 上？检验：令 s=1 得 (3,1,7)，是的，因此它们实际重合。
To test whether parallel lines are identical, substitute the coordinate of a known point from one line into the equation of the other line. This is a common IB exam question.
要判断两条平行直线是否重合，可将其中一条直线的已知点代入另一条直线的方程中。这是IB考试的常见题型。
9. Intersection and Skew Lines | 相交直线与异面直线
To find whether two non-parallel 3D lines intersect, solve the parametric equations simultaneously. Equate the x, y, and z components using different parameter names, then solve for the parameters and check consistency.
要判断两条不平行三维直线是否相交，可联立参数方程。使用不同参数名令 x、y、z 分量相等，解出参数并验证一致性。
For example, L₁: (x,y,z) = (1,2,3) + t(1,0,-1) and L₂: (x,y,z) = (3,0,2) + s(0,1,1). Equating gives:
例如，L₁: (x,y,z) = (1,2,3) + t(1,0,-1) 与 L₂: (x,y,z) = (3,0,2) + s(0,1,1)。令各分量相等：
1 + t = 3, 2 = s, 3 - t = 2 + s. From the first, t = 2; from the second, s = 2; the third gives 3 - 2 = 1, but 2 + 2 = 4. Contradiction, so the lines do not intersect.
1 + t = 3，2 = s，3 - t = 2 + s。由第一个得 t = 2；第二个得 s = 2；第三个给出 3 - 2 = 1，但 2 + 2 = 4。矛盾，因此两直线不相交。
In 3D, non-parallel lines that do not meet are called skew lines. This concept is fundamental in spatial geometry and often appears in IB Paper 3.
在三维空间中，不平行但不相交的直线称为异面直线。这一概念是空间几何的基础，常出现在IB Paper 3中。
10. Worked Example: Constructing a Line from a Point and Direction | 综合例题：由点和方向构建直线
Let us apply the methods to a typical IB-style problem. A line passes through point P(3, -1, 2) and is parallel to the line x - 24 = y + 5-2 = z7. Find the parametric equations of the line.
让我们用典型IB题来应用上述方法。一条直线经过点 P(3, -1, 2)，且平行于直线 x - 24 = y + 5-2 = z7。求该直线的参数方程。
Solution: The given line has direction vector d = (4, -2, 7). Since our line is parallel, it can use the same direction vector. With point P, the parameter equations are:
解：已知直线的方向向量为 d = (4, -2, 7)。因为所求直线平行于已知直线，所以可采用相同的方向向量。以点 P 为准，参数方程为：
x = 3 + 4t, y = -1 - 2t, z = 2 + 7t
We can also write the symmetric equations as x - 34 = y + 1-2 = z - 27.
对称方程为 x - 34 = y + 1-2 = z - 27。
Always remember: the direction vector of a line is not unique; any non-zero scalar multiple works. However, the equation structure remains standard, and you must carefully read the problem to decide whether the line passes through a given point or has a given direction.
永远记住：直线方向向量不唯一；任何非零标量倍均有效。但方程的结构保持标准，须仔细审题以确定直线经过给定点还是具有给定方向。
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