📚 Points in 3D Space | 三维空间中的点
Understanding points in three-dimensional space extends the coordinate geometry you know from the plane into a new dimension. This article explores how to describe, measure, and manipulate points in 3D, laying the foundation for vectors, surfaces, and spatial reasoning essential in IB Mathematics.
理解三维空间中的点能将你熟悉的平面坐标几何延伸至新的维度。本文探讨如何在三维空间中描述、测量和处理点，为向量、曲面以及 IB 数学所必需的空间推理奠定基础。
1. The 3D Coordinate System | 三维坐标系
A point in space is located using three mutually perpendicular axes: the x-axis, y-axis, and z-axis. These axes intersect at the origin O(0, 0, 0). The system follows the right-hand convention: if you point your right index finger along the positive x-axis and your middle finger along the positive y-axis, your thumb points along the positive z-axis.
空间中的点通过三个相互垂直的轴来确定：x 轴、y 轴和 z 轴。这些轴相交于原点 O(0, 0, 0)。坐标系遵循右手定则：若将右手食指指向正 x 轴、中指指向正 y 轴，则拇指指向正 z 轴。
The three axes divide space into eight octants. In the first octant, all coordinates are positive. Knowing the sign pattern in each octant helps you quickly identify a point's rough location without plotting it precisely.
三条轴将空间分为八个卦限。在第一卦限中，所有坐标均为正值。了解每个卦限中的正负号规律，有助于你无需精确绘图即可快速判断点的大致位置。
2. Coordinates of a Point | 点的坐标
A point P in three dimensions is written as an ordered triple (x, y, z). The x-coordinate measures the signed distance from the yz-plane, the y-coordinate from the xz-plane, and the z-coordinate from the xy-plane.
三维空间中的点 P 记作有序三元组 (x, y, z)。x 坐标表示该点到 yz 平面的有向距离，y 坐标表示到 xz 平面的有向距离，z 坐标表示到 xy 平面的有向距离。
For example, the point A(3, -2, 5) lies 3 units in front of the yz-plane, 2 units to the left of the xz-plane, and 5 units above the xy-plane. Understanding these projections is crucial when sketching 3D diagrams by hand.
例如，点 A(3, -2, 5) 位于 yz 平面前 3 个单位、xz 平面左侧 2 个单位、xy 平面上方 5 个单位处。理解这些投影对于手工绘制三维草图至关重要。
3. Distance Between Two Points | 两点间的距离
The distance d between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is a direct extension of the 2D distance formula. By applying the Pythagorean theorem twice, we obtain the distance formula in three dimensions.
两点 A(x₁, y₁, z₁) 与 B(x₂, y₂, z₂) 之间的距离 d 是二维距离公式的直接推广。通过两次应用勾股定理，可得三维空间中的距离公式。
d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
This formula is symmetric and always yields a non-negative result. It forms the basis for defining spheres and for solving geometric problems about triangles and tetrahedrons in space.
该公式具有对称性，结果始终非负。它为定义球体以及求解空间中三角形和四面体的几何问题奠定了基础。
4. Midpoint of a Segment | 线段的中点
The midpoint M of the segment joining A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is found by averaging the corresponding coordinates. It is the point that divides the segment into two equal lengths.
连接 A(x₁, y₁, z₁) 和 B(x₂, y₂, z₂) 的线段的中点 M 可以通过对相应坐标求平均数得到。该点将线段分成两条等长的部分。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
Midpoints appear frequently when constructing medians of triangles or when finding the centre of a parallelogram in 3D. With vectors, the midpoint can also be expressed as (a + b)/2, where a and b are position vectors.
在构造三角形的中线或寻找三维平行四边形中心时，中点经常出现。借助向量，中点也可表示为 (a + b)/2，其中 a 和 b 为位置向量。
5. Section Formula: Internal Division | 内分点的定比分点公式
If a point P divides the line segment from A(x₁, y₁, z₁) to B(x₂, y₂, z₂) internally in the ratio m : n, its coordinates can be found using the section formula. This generalises the midpoint as a special case.
若点 P 将线段 AB（从 A(x₁, y₁, z₁) 到 B(x₂, y₂, z₂)）以 m : n 内分，其坐标可通过定比分点公式求得。该公式将中点公式推广为一般形式。
P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n))
int. The formula is essential for problems involving centroids of triangles, where the centroid divides each median in the ratio 2:1 from vertex to midpoint.
当 m = n = 1 时，该公式退化为中点公式。对于涉及三角形重心的问题，该公式至关重要，因为重心将每条中线以 2:1 的比例从顶点分至中点。
6. Vectors and Position Vectors | 向量与位置向量
where i, j, k are the unit vectors along the axes. This allows us to treat points as vectors and use vector algebra to describe spatial relationships.
点 P(x, y, z) 可与其位置向量 OP = xi + yj + zk 关联，其中 i, j, k 为沿坐标轴的单位向量。这使得我们可以将点视为向量，并用向量代数描述空间关系。
₂ − x₁)i + (y₂ − y₁)j + (z₂ − z₁)k. Its magnitude equals the distance between the points, linking the geometric and algebraic viewpoints seamlessly.
从 A 到 B 的向量为 AB = (x₂ − x₁)i + (y₂ − y₁)j + (z₂ − z₁)k。其模等于两点间的距离，由此无缝连接了几何与代数视角。
7. Equation of a Sphere | 球面方程
Just as the set of points at a fixed distance from a centre defines a circle in 2D, in 3D it defines a sphere. A sphere with centre C(a, b, c) and radius r has a simple standard equation derived directly from the distance formula.
正如与定点距离为定值的点集在二维中定义圆一样，在三维中它定义了一个球面。以 C(a, b, c) 为心、半径为 r 的球面具有一个由距离公式直接导出的简洁标准方程。
(x − a)² + (y − b)² + (z − c)² = r²
s you to recover the centre coordinates and radius, a skill frequently tested in IB exams.
展开此方程可得一般形式 x² + y² + z² + 2gx + 2fy + 2hz + d = 0。通过配方，你可以重新获得球心坐标与半径，这是 IB 考试中常考的技能。
8. Translating Points Using Vectors | 利用向量平移点
ing a translation vector v = pi + qj + rk to a point A(x, y, z) shifts it to a new point A' + p, y + q, z + r). This operation preserves distances and shapes, making it an isometry.
给点 A(x, y, z) 加上平移向量 v = pi + qj + rk，可将其移动至新点 A'(x + p, y + q, z + r)。该操作保持距离和形状不变，因此是一种等距变换。
Translation is fundamental in transforming geometric figures. For instance, the image of a line segment under translation has the same length and direction, and the midpoint translates accordingly.
平移是几何图形变换的基础。例如，线段在平移下的像具有相同的长度和方向，中点也相应地发生平移。
9. Collinearity of Three Points | 三点共线
Three points A, B, and C are collinear if the vectors AB and BC are parallel. This is equivalent to checking whether the determinant formed by the direction vectors is zero or whether one point satisfies the section formula with a specific ratio.
若向量 AB 与 BC 平行，则三点 A、B、C 共线。这等价于检查由方向向量构成的矩阵行列式是否为零，或者某一点是否满足具有特定比例的内分点公式。
A practical test is to see if the coordinates of C can be expressed as A + t(B − A) for some scalar t. If such a t exists and lies between 0 and 1, C is on the segment AB; otherwise, it lies on the extension.
一种实用的检验方法是观察能否将 C 的坐标表示为 A + t(B − A)，其中 t 为某个标量。若存在这样的 t 且在 0 到 1 之间，则 C 在线段 AB 上；否则它位于延长线上。
10. Symmetry and Reflections | 对称与反射
Reflecting a point across the xy-plane changes the sign of its z-coordinate: (x, y, z) becomes (x, y, −z). Similar rules apply for reflections across the xz- or yz-plane, and across the origin.
将点关于 xy 平面对称会改变其 z 坐标的符号：(x, y, z) 变为 (x, y, −z)。关于 xz 平面或 yz 平面的反射以及关于原点的反射也遵循类似的规则。
Reflection across a general plane or line is more advanced but can be handled with vector methods. In IB Mathematics, you are expected to recognise these basic symmetries and use them to find coordinates quickly.
关于一般平面或直线的反射更为深入，但可通过向量方法处理。在 IB 数学中，你应能识别这些基本对称关系，并利用它们快速求得坐标。
11. Practical Problem: Finding a Missing Vertex | 实际应用：求缺失的顶点
Given three vertices of a parallelogram A(1, 2, 3), B(4, 1, 5), and C(6, 3, 8), you can find the fourth vertex D by using the property that the diagonals bisect each other. The midpoint of AC must equal the midpoint of BD.
已知平行四边形三个顶点 A(1, 2, 3)、B(4, 1, 5) 和 C(6, 3, 8)，可利用对角线互相平分的性质求出第四个顶点 D。AC 的中点必须等于 BD 的中点。
. Solving gives D(3, 4, 6). This method illustrates the tight connection between coordinate geometry and vector logic.
建立方程：(1+6)/2 = (4+x)/2，(2+3)/2 = (1+y)/2，(3+8)/2 = (5+z)/2。求解得 D(3, 4, 6)。此方法展示了坐标几何与向量逻辑之间的紧密联系。
12. Key Concepts Summary | 核心概念总结
Mastering points in 3D space boils down to internalising the distance formula, midpoint, section formula, and the link to vectors. These tools allow you to tackle a wide range of spatial geometry problems, from finding centres of spheres to proving collinearity.
掌握三维空间中的点归根结底在于内化距离公式、中点、定比分点公式以及与向量的联系。这些工具使你能够处理从求球心到证明共线等广泛的空间几何问题。
Always draw a rough sketch, label projections, and verify coordinates with simple checks. With practice, 3D coordinate geometry becomes as intuitive as its 2D counterpart, providing a powerful language for scientific and engineering applications.
务必绘制草图，标注投影，并通过简单检验验证坐标。经过练习，三维坐标几何将变得与二维一样直观，为科学和工程应用提供一种有力的语言。
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