📚 Essential Guide to 3D Coordinate Geometry | 三维坐标几何核心知识解析
Three-dimensional coordinate geometry is a cornerstone of IB Mathematics Analysis and Approaches Higher Level, appearing in both Paper 2 and Paper 3 questions. This guide consolidates the essential definitions, formulas, and problem-solving techniques you need for the exam.
三维坐标几何是 IB 数学分析与方法（AA）高级课程的核心内容，在 Paper 2 和 Paper 3 中频繁出现。本指南将整合你在考试中所需的关键定义、公式与解题技巧。
1. The Three-Dimensional Cartesian Coordinate System | 三维笛卡尔坐标系
In 3D, a point P is defined by an ordered triple (x, y, z), where x, y, and z are the signed distances from the point to the three mutually perpendicular coordinate planes (yz-plane, xz-plane, and xy-plane respectively). The coordinate axes intersect at the origin O(0, 0, 0).
在三维空间中，点 P 由有序三元组 (x, y, z) 确定，其中 x、y、z 分别是该点到三个互相垂直的坐标平面（yz 平面、xz 平面、xy 平面）的有向距离。三条坐标轴相交于原点 O(0, 0, 0)。
By convention, the axes follow the right-hand rule: if the index finger of the right hand points along the positive x-axis and the middle finger bends toward the positive y-axis, the thumb points along the positive z-axis. The three coordinate planes divide space into eight octants.
按照惯例，坐标轴遵循右手定则：右手的食指指向 x 轴正方向，中指弯向 y 轴正方向时，大拇指指向 z 轴正方向。三个坐标平面将空间分成八个卦限。
2. Distance and Midpoint Formulas | 距离与中点公式
Given two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), the distance between them is found by applying Pythagoras’ theorem twice. The formula is an extension of the 2D distance formula into the third dimension.
已知两点 P₁(x₁, y₁, z₁) 和 P₂(x₂, y₂, z₂)，它们之间的距离可以通过两次应用勾股定理求得。该公式是二维距离公式向三维空间的推广。
|P₁P₂| = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²]
The midpoint M of segment P₁P₂ has coordinates that are the arithmetic mean of the coordinates of the endpoints:
线段 P₁P₂ 的中点 M 的坐标是两端点坐标的算术平均值：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
More generally, the point that divides the segment P₁P₂ in the ratio m : n can be obtained using the section formula. This is especially useful in coordinate geometry problems involving medians of triangles in space.
更一般地，按 m : n 的比值分割线段 P₁P₂ 的点可用定比分点公式求得。这在涉及空间三角形中线问题的坐标几何题中尤为实用。
3. Vectors in Three Dimensions | 三维向量基础
A vector in 3D has both magnitude and direction. The position vector of a point A(x, y, z) relative to the origin is written as a = (x, y, z). The vector from A to B is obtained by subtracting position vectors: AB = b – a.
三维向量具有大小和方向。点 A(x, y, z) 相对于原点的位置向量记为 a = (x, y, z)。从 A 到 B 的向量通过位置向量相减得到：AB = b – a。
The magnitude (or length) of vector u = (u₁, u₂, u₃) is:
向量 u = (u₁, u₂, u₃) 的模（长度）为：
|u| = √(u₁² + u₂² + u₃²)
Key vector operations in 3D include scalar multiplication ku = (ku₁, ku₂, ku₃) and vector addition/subtraction performed component-wise. A unit vector in the direction of u is û = u / |u|. The standard basis vectors are i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1).
三维空间中的关键向量运算包括标量乘法 ku = (ku₁, ku₂, ku₃) 以及逐分量进行的向量加减法。u 方向上的单位向量为 û = u / |u|。标准基向量为 i = (1, 0, 0)、j = (0, 1, 0)、k = (0, 0, 1)。
4. The Dot Product | 向量点积
The dot product (scalar product) of two vectors u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃) is defined algebraically as the sum of the products of corresponding components:
两个向量 u = (u₁, u₂, u₃) 与 v = (v₁, v₂, v₃) 的点积（数量积）在代数上定义为对应分量乘积之和：
u ⋅ v = u₁v₁ + u₂v
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