IB Mathematics: Equations of Lines in 2D and 3D Space | IB数学：二维与三维空间中的直线方程

📚 IB Mathematics: Equations of Lines in 2D and 3D Space | IB数学：二维与三维空间中的直线方程

For IB students, understanding straight lines in two and three dimensions is fundamental for analytic geometry and vector geometry. In 2D, lines are simply expressed as linear equations; in 3D, the parametric and vector forms become essential. This article summarises the key formulas, methods and exam techniques.

对于IB学生而言，理解二维和三维空间中的直线方程是解析几何和向量几何的基础。在二维中，直线可以用线性方程简单表达；而在三维中，参数方程和向量方程形式则至关重要。本文将总结关键公式、方法和考试技巧。

1. Equations of Lines in 2D | 二维空间中的直线方程

There are several common forms for a line in the Cartesian plane. The gradient form is y = mx + c, where m is the gradient and c is the y-intercept. The general form is ax + by = d, where (a, b) is a normal vector to the line. Another useful form is the point-slope equation y – y₁ = m(x – x₁), which uses a point P₁(x₁, y₁) on the line and the gradient m.

笛卡尔平面中直线有若干种常见形式。斜截式为 y = mx + c，其中 m 是斜率，c 是 y 截距。一般式为 ax + by = d，其中 (a, b) 是直线的法向量。另一种常用形式是点斜式 y – y₁ = m(x – x₁)，它使用直线上一点 P₁(x₁, y₁) 和斜率 m。

Given two points A(x₁, y₁) and B(x₂, y₂), the gradient is m = (y₂ – y₁)/(x₂ – x₁). The equation of the line can then be written using the point-slope form. In IB exams, you may also need to use the vector form: r = a + t b, where a is the position vector of a known point on the line and b is a non-zero direction vector.

若已知两点 A(x₁, y₁) 和 B(x₂, y₂)，则斜率 m = (y₂ – y₁)/(x₂ – x₁)。随后可用点斜式写出直线方程。在IB考试中，还可能需要使用向量形式：r = a + t b，其中 a 是直线上已知点的位置向量，b 是非零方向向量。

In 2D, the parametric form can be written as x = x₀ + p t and y = y₀ + q t, where the direction vector is (p, q). Eliminating t gives the familiar Cartesian equation. This same idea extends naturally into three dimensions.

在二维中，参数形式可写为 x = x₀ + p t 和 y = y₀ + q t，其中方向向量为 (p, q)。消去 t 得到熟知的笛卡尔方程。这一思想可以自然推广到三维空间。

2. Vector Equation of a Line in 3D | 三维空间中的直线向量方程

In 3D, a line is determined by a fixed point A, with position vector a, and a direction vector b. The vector equation of the line is r = a + t b, where t is a real parameter and each point on the line corresponds to exactly one value of t. As t varies over all real numbers, r traces out the entire infinite line.

在三维空间中，一条直线可由固定点 A（位置向量为 a）和方向向量 b 确定。直线的向量方程为 r = a + t b，其中 t 是实数参数，直线上的每一点都唯一对应一个 t 值。当 t 取遍所有实数时，r 描出整条无限直线。

r = a + t b, t ∈ ℝ

If a = (x₀, y₀, z₀) and b = (l, m, n), then the parametric equations of the line are obtained by comparing components:

若 a = (x₀, y₀, z₀)，b = (l, m, n)，则比较分量可得直线的

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