📚 A-Level Mathematics: 3D Coordinate Systems Explained | A-Level 数学：三维空间坐标系详解
In A-Level Mathematics, the move from 2D to 3D coordinate geometry is an important step that connects algebra, vectors, and spatial reasoning. This article explains the core ideas you need: axes, coordinates, distance, midpoints, vectors, lines, planes, and angles in three-dimensional space.
在 A-Level 数学中，从二维平面到三维空间坐标几何的过渡，是将代数、向量与空间思维联系起来的重要环节。本文系统讲解三维坐标系中的核心概念：坐标轴、点的坐标、距离、中点、向量、直线、平面以及空间夹角。
1. The Three Axes | 坐标轴
axis, the y-axis, and the z-axis. These axes meet at the origin, written as O = (0, 0, 0).
在三维坐标几何中，我们用三条互相垂直的坐标轴来确定点的位置，它们分别是 x 轴、y 轴和 z 轴。这三条轴相交于原点，记作 O = (0, 0, 0)。
The x-axis and y-axis are usually drawn on the horizontal plane, while the z-axis is often shown as the vertical direction in exam diagrams.
x 轴和 y 轴通常位于水平平面内，而 z 轴在考试图形中常被画成竖直方向。
The positive directions are normally chosen using the right-hand rule, which gives a consistent orientation for all three axes.
三个轴的正方向通常由右手定则确定，以保证整个坐标系的方向一致。
Every point in 3D space is represented by an ordered triple (x, y, z).
三维空间中的每一个点都用有序三元组 (x, y, z) 表示。
2. Coordinate Planes | 坐标平面
The three coordinate axes define three coordinate planes, and each plane is formed by setting one coordinate equal to zero.
三条坐标轴共同确定出三个坐标平面，每个坐标平面都由令某一个坐标等于 0 得到。
he xy-plane contains the x-axis and the y-axis, and it is described by the equation z = 0.
xy 平面包含 x 轴和 y 轴，其方程为 z = 0。
he yz-plane contains the y-axis and the z-axis, and it is described by the equation x = 0.
yz 平面包含 y 轴和 z 轴，其方程为 x = 0。
he xz-plane contains the x-axis and the z-axis, and it is described by the equation y = 0.
xz 平面包含 x 轴和 z 轴，其方程为 y = 0。
These three planes divide space into eight regions, often called octants.
这三个坐标平面把空间分成八个区域，通常称为卦限。
3. Coordinates of a Point | 点的坐标
To reach a point P(x, y, z) from the origin, move x units along the x-axis, then y units along the y-axis, then z units along the z-axis.
从原点出发到达点 P(x, y, z) 时，我们沿 x 轴移动 x 个单位，再沿 y 轴移动 y 个单位，最后沿 z 轴移动 z 个单位。
The x-coordinate represents the signed distance from the yz-plane.
x 坐标表示点到 yz 平面的有向距离。
The y-coordinate represents the signed distance from the xz-plane.
y 坐标表示点到 xz 平面的有向距离。
The z-coordinate represents the signed distance from the xy-plane.
z 坐标表示点到 xy 平面的有向距离。
The origin has coordinates (0, 0, 0), and every point in space can be expressed uniquely in this form.
原点的坐标是 (0, 0, 0)，空间中每一个点都可以唯一地写成这种形式。
4. Distance Between Two Points | 两点间的距离
If two points are given by P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), the distance between them is obtained by applying Pythagoras' theorem to all three dimensions.
如果两个点的坐标分别为 P₁(x₁, y₁, z₁) 和 P₂(x₂, y₂, z₂)，则两点间的距离可以通过将勾股定理推广到三维空间来得到。
d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
This formula is the 3D version of the familiar 2D distance formula.
这个公式是熟悉的三维版距离公式。
If the point P is the origin, then the distance to P(x, y, z) is simply √x² + y² + z².
如果点 P 是原点，那么点 P(x, y, z) 到原点的距离就是 √x² + y² + z²。
You may also see this written as d = |P₁P₂|, meaning the length of the vector from P₁ to P₂.
这个距离也常被记作 d = |P₁P₂|，表示从 P₁ 到 P₂ 的向量长度。
5. Midpoint of a Segment | 线段中点
The midpoint M of the segment joining P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) is found by averaging the coordinates of the two endpoints.
连接两点 P₁(x₁, y₁, z₁) 和 P₂(x₂, y₂, z₂) 的线段中点 M，可以通过求两端点坐标的平均值得到。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
This is exactly the same pattern as the midpoint formula in 2D, extended with a z-term.
这与二维中点公式的模式完全相同，只是增加了一个 z 分量。
It is useful when working with medians of triangles, centres of edges, and other geometric constructions in space.
在计算空间中三角形的中线、棱的中点以及其他几何构造时，这个公式非常有用。
6. Vectors in 3D | 三维向量
In three dimensions, a vector can be written using the unit vectors i, j, and k along the x-, y-, and z-axes respectively.
在三维空间中，一个向量可以分别沿 x 轴、y 轴和 z 轴的单位向量 i、j、k 来表示。
A position vector r = xi + yj + zk corresponds to the point (x, y, z).
位置向量 r = xi + yj + zk 对应点 (x, y, z)。
The magnitude of a vector r = xi + yj + zk is given by:
向量 r = xi + yj + zk 的模长为：
|r| = √x² + y² + z²
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