📚 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
In three-dimensional coordinate geometry, a point is described using three mutually perpendicular axes called the x-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)。
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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。
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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。
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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。
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The 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。
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The 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。
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The 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。
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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。
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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。
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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。
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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。
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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)
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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。
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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)。
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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)
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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。
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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。
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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)。
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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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