📚 3D Coordinates | 三维坐标
The concept of 3D coordinates extends the familiar 2D Cartesian plane into a third dimension, allowing us to describe the position of points in space. In A-Level Mathematics (Edexcel), understanding 3D coordinates is essential for working with vectors, lines, planes, and solving geometrical problems in three dimensions. This article covers the fundamentals from plotting points to the equations of lines and planes, with clear explanations and key formulae.
三维坐标的概念将我们熟悉的二维笛卡尔平面扩展到第三个维度，使我们能够描述空间中的点的位置。在A-Level数学（Edexcel）中，理解三维坐标对于处理向量、直线、平面以及解决三维几何问题至关重要。本文涵盖从绘制点到直线和平面的方程等基础知识，并提供清晰的解释和关键公式。
1. The 3D Cartesian Coordinate System | 三维笛卡尔坐标系
In 3D, we use three mutually perpendicular axes: the x-axis, y-axis, and z-axis, all intersecting at the origin O(0,0,0). The axes form a right-handed system: if you point your right-hand fingers along the positive x-axis and curl them towards the positive y-axis, your thumb points along the positive z-axis. Points are specified by ordered triples (x, y, z).
在三维中，我们使用三条相互垂直的坐标轴：x轴、y轴和z轴，它们相交于原点O(0,0,0)。这些轴构成右手系：如果你将右手手指指向x轴正方向，并向y轴正方向弯曲，那么拇指所指的方向就是z轴正方向。点用有序三元组(x, y, z)表示。
2. Plotting and Visualising Points | 绘制与想象点
To plot a point P(a, b, c), move a units along the x-axis, then b units parallel to the y-axis, and finally c units parallel to the z-axis. Visualising 3D points on paper requires imagining the z-axis coming out of the page. Many exam questions provide diagrams, but you can also project points onto coordinate planes to analyse their positions more easily.
要绘制点P(a, b, c)，先沿x轴移动a个单位，然后平行于y轴移动b个单位，最后平行于z轴移动c个单位。在纸上想象三维点需要设想z轴从纸面伸出来。许多考题会提供示意图，但你也可以将点投影到坐标平面上，以便更轻松地分析其位置。
3. Distance Formula in 3D | 三维距离公式
The distance between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is an extension of Pythagoras’ theorem:
AB = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
This formula is fundamental for calculating lengths of line segments in space. It can also be used to derive the equation of a sphere: (x − a)² + (y − b)² + (z − c)² = r², with centre (a,b,c) and radius r. Make sure to apply the formula carefully, especially when coordinates contain negative values.
点A(x₁, y₁, z₁)和B(x₂, y₂, z₂)之间的距离是勾股定理的延伸：
AB = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
该公式是计算空间中线段长度的基础。它也可用于推导球面方程：(x − a)² + (y − b)² + (z − c)² = r²，其中球心为(a,b,c)，半径为r。应用公式时务必仔细，尤其在坐标含有负值时。
4. Midpoint Formula | 中点公式
The midpoint M of the line segment joining A(x₁, y₁, z₁) and B(x₂, y₂, z₂) has coordinates:
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 , (z₁ + z₂)/2 )
This is simply the average of the respective coordinates. It is useful for finding the centre of a line segment or for symmetry problems, such as determining the reflection of a point across a midpoint.
连接点A(x₁, y₁, z₁)和B(x₂, y₂, z₂)的线段中点M的坐标为：
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 , (z₁ + z₂)/2 )
这只是相应坐标的平均值。在寻找线段中心或对称性问题（如求点关于中点的反射点）中很有用。
5. Vectors in 3D | 三维向量
A vector in 3D can be expressed in component form as a = xi + yj + zk, where i, j, k are the unit vectors along the x, y, z axes respectively. The position vector of a point P(x, y, z) is r = xi + yj + zk. Vector operations such as addition, subtraction, and scalar multiplication are performed component-wise. The magnitude (length) of vector a is |a| = √(x² + y² + z²).
三维向量可以用分量形式表示为 a = xi + yj + zk，其中 i, j, k 分别是沿x、y、z轴方向的单位向量。点P(x, y, z)的位置向量为 r = xi + yj + zk。向量的加法、减法和数乘等运算都是按分量进行的。向量 a 的大小（模）为 |a| = √(x² + y² + z²)。
6. The Equation of a Straight Line in 3D | 三维空间直线方程
A line passing through a point A with position vector a and parallel to a direction vector b can be written in vector parametric form:
r = a + λb, λ ∈ ℝ
In component form, if a = (x₀, y₀, z₀) and b = (l, m, n), then the Cartesian equations are:
(x − x₀)/l = (y − y₀)/m = (z − z₀)/n (= λ)
provided l, m, n are non-zero. If any direction ratio is zero, the corresponding coordinate remains constant; for instance, if l = 0, then x = x₀ while the other ratios hold. This is the symmetric form of the line.
过点A（位置向量为 a）且平行于方向向量 b 的直线可以用向量参数方程表示为：
r = a + λb, λ ∈ ℝ
用分量形式，若 a = (x₀, y₀, z₀)，b = (l, m, n)，则笛卡尔方程为：
(x − x₀)/l = (y − y₀)/m = (z − z₀)/n (= λ)
前提是l, m, n均不为零。若某个方向比为零，则对应的坐标保持不变；例如若l = 0，则x = x₀，而其他比例式仍然成立。这就是直线的对称式方程。
7. The Equation of a Plane | 平面方程
A plane can be defined using a normal vector n = (a, b, c) perpendicular to the plane. The scalar product form is r · n = a · n where a is the position vector of a known point on the plane. Expanding gives the Cartesian equation:
ax + by + cz = d
where d = a·x₀ + b·y₀ + c·z₀ for a point (x₀, y₀, z₀) on the plane. Alternatively, a plane can be expressed in parametric form using two non-parallel direction vectors and a point: r = a + λu + μv.
平面可以用与该平面垂直的法向量 n = (a, b, c) 来定义。点积形式为 r · n = a · n，其中 a 是平面上已知点的位置向量。展开得到笛卡尔方程：
ax + by + cz = d
其中对于平面上一点 (x₀, y₀, z₀)，d = a·x₀ + b·y₀ + c·z₀。平面也可以用两个不平行方向向量和一个点的参数形式表示：r = a + λu + μv。
8. Angle Between Two Lines | 两线夹角
If two lines have direction vectors b and d, the acute angle θ between them satisfies:
cos θ = |b · d| / (|b| |d|)
The absolute value guarantees the acute angle. This is directly analogous to the 2D formula, extended to three components. Always use the direction vectors when applying this formula – do not use position vectors of points on the lines.
如果两条直线的方向向量分别为 b 和 d，它们之间的锐角 θ 满足：
cos θ = |b · d| / (|b| |d|)
取绝对值是为了保证锐角。该公式是二维公式的直接推广，扩展到三个分量。应用此公式时请始终使用方向向量——不要使用直线上点的位置向量。
9. Angle Between a Line and a Plane | 线与平面的夹角
The angle between a line (direction vector b) and a plane (normal vector n) is the complement of the angle between b and n. If φ is the angle between the line and the plane, then:
sin φ = |b · n| / (|b| |n|)
This makes sense: when the line is parallel to the plane, b · n = 0 ⇒ φ = 0°; when the line is perpendicular to the plane, b is parallel to n ⇒ φ = 90°.
直线（方向向量 b）与平面（法向量 n）之间的夹角是 b 与 n 夹角的余角。若直线与平面的夹角为 φ，则：
sin φ = |b · n| / (|b| |n|)
这合乎情理：当直线与平面平行时，b · n = 0 ⇒ φ = 0°；当直线与平面垂直时，b 平行于 n ⇒ φ = 90°。
10. Distance from a Point to a Plane | 点到平面的距离
The perpendicular distance
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