📚 8H Angles and Intersections Between Lines and Planes | 8H 线与平面的夹角与交点
Mastering the angles and intersections of lines and planes is a cornerstone of the IB Mathematics vector geometry unit. This article provides a thorough guide to computing angles between lines, between a line and a plane, and between two planes, as well as finding points of intersection, lines of intersection, and related distances. Each section pairs an English explanation with a Chinese equivalent, equipped with essential formulas, worked examples and exam-focused warnings.
掌握直线与平面的夹角和交点是 IB 数学向量几何单元的基础。本文提供计算线线角、线面角、面面角的详尽指南，以及求交点、交线、相关距离与垂足的方法。每个部分均配有中英文解释、核心公式、例题与考试提醒，帮助你在 Paper 1 与 Paper 2 的向量题中稳定拿分。
1. Vector Review and Notation | 向量复习与符号
In three-dimensional geometry, a vector is usually written in component form as a = (a₁, a₂, a₃), where a₁, a₂, a₃ are real numbers. The magnitude, or length, of a is given by the square root of the sum of the squares of its components. The dot product of two vectors is a scalar, is symmetric, and satisfies a·a = |a|², which is the key identity used to compute angles.
在三维几何中，向量通常写为分量形式 a = (a₁, a₂, a₃)，其中 a₁、a₂、a₃ 为实数。向量 a 的模（长度）等于各分量平方和的开方。两个向量的点积是标量，满足交换律，并且有 a·a = |a|²，这是计算夹角时最重要的恒等式。
|a| = √a₁² + a₂² + a₃² a·b = a₁b₁ + a₂b₂ + a₃b₃
The cross product a × b produces a vector perpendicular to both a and b, and it is the standard tool for building a normal vector from two direction vectors. Throughout this article, a line is described by its direction vector d, and a plane by its normal vector n een two vectors satisfies the cosine formula below, where 0 ≤ θ ≤ π (that is, 0° to 180°).
叉积 a × b 生成一个同时垂直于 a 与 b 的向量，是由两个方向向量构造法向量的标准工具。本文中，直线用方向向量 d 描述，平面用法向量 n 描述。两向量夹角 θ 满足下面的余弦公式，其中 0 ≤ θ ≤ π（即 0° 到 180°）。
cos θ = (a·b) ÷ (|a| |b|) , 0 ≤ θ ≤ π
A unit vector in the direction of a is written as a with a hat, equal to a divided by |a|. This normalisation appears constantly when projecting one vector onto another, which is exactly what finding an angle or a perpendicular foot requires.
与 a 同方向的单位向量记作 a 上加帽，等于 a 除以 |a|。这种单位化在把一个向量投影到另一个向量上时反复出现，而求夹角或求垂足本质上就是做投影。
2. Equations of Lines in 3D | 空间直线的方程
A line in 3D is most conveniently expressed in vector parametric form: r = a + λd, where a is the position vector of a known point on the line, d is the direction vector, and λ is a real parameter. As λ varies over all real numbers, the point r traces the entire line in both directions.
三维空间中的直线最方便的表示是向量参数式：r = a + λd，其中 a 为直线上已知点的位置向量，d 为方向向量，λ 为实数参数。当 λ 取遍全体实数时，点 r 会描出整条直线（双向无限延伸）。
r = a + λd ⟺ (x − x₀)/d₁ = (y − y₀)/d₂ = (z − z₀)/d₃
The symmetric, or Cartesian, form on the right is obtained by eliminating λ from the three component equations. If two points A and B on the line are known, the direction vector is simply d = b − a −2, 3) with direction (2, 1, −4) has symmetric form (x − 1)/2 = (y + 2)/1 = (z − 3)/(−4).
右端的对称式（笛卡尔式）由三个分量方程消去 λ 得到。若已知直线上两点 A 与 B，则方向向量直接取 d = b − a。例如过点 (1, −2, 3) 且方向为 (2, 1, −4) 的直线，对称式为 (x − 1)/2 = (y + 2)/1 = (z − 3)/(−4)。
- If one component of d ot place it in a denominator; write that coordinate as a separate equation such as x = x₀.
若 d 的某个分量为零，不要把它写进分母，而应把该坐标单独写成 x = x₀ 这类方程。 - Any non-zero scalar multiple of d describes the same line, so direction vectors are never unique.
d 的任意非零数乘仍表示同一条直线，因此方向向量并不唯一。 - Understanding these three representations is fundamental for finding intersections later in this chapter.
掌握这三种表示形式，是本章后续求交点问题的基础。
3. Equations of Planes in 3D | 空间平面的方程
A plane can be written in scalar form ax + by + cz = d with normal vector n = (a, b, c). The vector equation of the same plane is r·n = a·n, where a is the position vector of a known point on the plane. Alternatively, a plane can be defined by a point and two non-parallel direction vectors, giving the parametric form r = a + λu + μv.
平面可以写成标量形式 ax + by + cz = d，其法向量为 n = (a, b, c)。同一平面的向量方程是 r·n = a·n，其中 a 是平面上已知点的位置向量。此外，平面也可由一点和两个不平行的方向向量确定，即参数式 r = a + λu + μv。
ax + by + cz = d ⟺ r·n = a·n ⟺ r = a + λu + μv
Given three non-collinear points A, B and C, a normal vector is obtained from the cross product n = (b − a) × (c − a gh (1, 2, 3) with normal (2, −1, 4) is 2x − y + 4z = 2 − 2 + 12, that is 2x − y + 4z = 12.
已知不共线的三点 A、B、C，可用叉积求法向量 n = (b − a) ×
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