📚 3D Coordinates of Points in Space | IB数学：三维空间中的点坐标
In IB Mathematics, working with points in three-dimensional space is a foundational skill that extends the familiar 2D coordinate system into a third dimension. This article covers how to locate points, calculate distances and midpoints, and interpret geometrical relationships in 3D, all while following the IB syllabus closely.
在IB数学中，处理三维空间中的点是一项基础技能，它将熟悉的二维坐标系扩展到第三个维度。本文全面讲解如何确定点的位置、计算距离和中点，并解读三维空间中的几何关系，内容严格按照IB教学大纲编写。
1. The 3D Coordinate System | 三维坐标系
A three-dimensional coordinate system consists of three mutually perpendicular axes: the x-axis, the y-axis, and the z-axis. These axes intersect at a point called the origin, denoted by O(0, 0, 0). The position of any point P in space is given by an ordered triple (x, y, z), where x is the distance along the x-axis, y is the distance along the y-axis, and z is the height above or below the xy-plane.
三维坐标系由三条相互垂直的轴组成：x轴、y轴和z轴。这些轴交于一点，称为原点，记作O(0, 0, 0)。空间中任意点P的位置由有序三元组(x, y, z)表示，其中x是沿x轴的距离，y是沿y轴的距离，z是相对于xy平面的高度。
In IB Mathematics, you will often need to sketch points by drawing a 3D axis system. A good habit is to draw the x-axis coming toward you, the y-axis to the right, and the z-axis pointing upward—this is the standard right-handed coordinate system used in most textbooks.
在IB数学中，你经常需要绘制3D坐标系来标出点。一个好习惯是：x轴朝向读者，y轴朝右，z轴朝上——这是大多数教材中使用的标准右手坐标系。
2. Plotting a Point in Space | 在空间中确定点的位置
To plot a point P(x, y, z), start at the origin. Move x units along the x-axis, then y units parallel to the y-axis, and finally z units parallel to the z-axis. The order of movement does not matter; the final position is the same regardless of the path you take, as long as you respect the signed distances.
要绘制点P(x, y, z)，从原点出发。先沿x轴移动x个单位，然后沿平行于y轴的方向移动y个单位，最后沿平行于z轴的方向移动z个单位。移动的先后顺序不影响最终位置，只要保持有向距离的正确性，无论走哪条路径，得到的点都一样。
- If z = 0, the point lies in the xy-plane.
- If y = 0, the point lies in the xz-plane.
- If x = 0, the point lies in the yz-plane.
- If all coordinates are positive, the point is in the first octant.
- 如果z = 0，则该点位于xy平面内。
- 如果y = 0，则该点位于xz平面内。
- 如果x = 0，则该点位于yz平面内。
- 如果所有坐标都为正，则该点位于第一卦限。
3. Distance Between Two Points | 两点间的距离
Given two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), the distance d between them is found using the 3D distance formula. This formula is derived from the Pythagorean theorem applied twice: once in the xy-plane and once vertically.
给定两个点P₁(x₁, y₁, z₁)和P₂(x₂, y₂, z₂)，它们之间的距离d由三维距离公式计算。这个公式由勾股定理两次运用推导而来：一次在xy平面内，一次在垂直方向。
d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
This is one of the most important formulas in the 3D coordinate geometry chapter. You should understand its derivation rather than merely memorise it, because IB exam questions often ask you to apply it in unfamiliar contexts such as vector displacement or geometric shapes.
这是三维坐标几何章节中最重要的公式之一。你应该理解它的推导过程，而不是仅仅背下来，因为IB考试题经常要求你在陌生情境中应用它，例如向量位移或几何形状问题。
4. Worked Example: Distance Calculation | 例题演练：距离计算
Let’s calculate the distance between A(1, 2, 3) and B(4, −2, 7). Apply the formula directly:
让我们计算点A(1, 2, 3)与点B(4, −2, 7)之间的距离。直接套用公式：
d = √[(4 − 1)² + (−2 − 2)² + (7 − 3)²] = √[3² + (−4)² + 4²] = √[9 + 16 + 16] = √41
Therefore, the distance between A and B is √41 units. In IB exams, you may be asked to leave your answer in surd form, or to round to a required degree of accuracy, such as 3 significant figures (here, 6.40). Always read the question carefully.
因此，点A与点B之间的距离为√41个单位。在IB考试中，可能要求你保留根号形式，或按指定精度四舍五入，例如保留3位有效数字（此处为6.40）。务必仔细阅读题目要求。
5. Midpoint in 3D | 三维空间中的中点
The midpoint M of a segment joining P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) is simply the average of the corresponding coordinates. In 3D, the midpoint formula extends naturally from the 2D version:
连接P₁(x₁, y₁, z₁)与P₂(x₂, y₂, z₂)的线段的中点M，就是各对应坐标的平均值。在三维空间中，中点公式由二维版本自然推广而来：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
This formula is straightforward, but it has many applications. You may need to use it to find the center of a cube, the centroid of a triangle in space, or the intersection point of diagonals in a parallelepiped.
这个公式很简单，但应用非常广泛。你可能需要用它来找到立方体的中心、空间三角形的重心，或平行六面体对角线的交点。
6. Vectors and Points | 向量与点的关系
A point P(x, y, z) can be represented by its position vector, written as r = xi + yj + zk or as a column vector (x, y, z)ᵀ. The displacement vector from point A to point B is obtained by subtracting coordinates:
点P(x, y, z)可以用其位置向量表示，写作 r = xi + yj + zk 或列向量(x, y, z)ᵀ。从点A到点B的位移向量通过坐标相减得到：
→AB = OB − OA = (x₂ − x₁, y₂ − y₁, z₂ − z₁)
The magnitude of this vector is exactly the distance between A and B. This connection between coordinate geometry and vectors is frequently examined in IB Paper 1 and Paper 2.
这个向量的模长恰好等于A与B之间的距离。坐标几何与向量之间的联系是IB卷一和卷二中的常考内容。
7. Coordinates of Points on a Line Segment | 线段上点的坐标
A point dividing the segment AB in the ratio m : n can be found using the section formula. In 3D, this works coordinate by coordinate, just as in 2D. If P divides AB internally in the ratio m : n, then:
按比例m : n分割线段AB的点可以使用定比分点公式求得。在三维空间中，这个公式像二维中一样按坐标分别计算。如果P内分AB，比例为m : n，则：
P = ((nx₁ + mx₂)/(m + n), (ny₁ + my₂)/(m + n), (nz₁ + mz₂)/(m + n))
Note the pattern: the coordinate of A is multiplied by n (the ratio associated with the other end), and the coordinate of B is multiplied by m. A common mistake in IB exams is to swap these multipliers—always check that the midpoint case with m = n = 1 correctly gives the average.
注意这个模式：A的坐标乘以n（与另一端相关的比例），B的坐标乘以m。IB考试中常见的错误是交换这两个乘数——记得检查当m = n = 1时，该公式正确给出平均值的特例。
8. Equation of a Sphere | 球的方程
A sphere in 3D space is the set of all points that are a fixed distance r (the radius) from a fixed point C(a, b, c) (the center). The standard equation is:
三维空间中的球面是到定点C(a, b, c)（球心）距离恒为r（半径）的所有点的集合。其标准方程为：
(x − a)² + (y − b)² + (z − c)² = r²
For example, a sphere centered at the origin with radius 5 has the equation x² + y² + z² = 25. To determine whether a given point lies on, inside, or outside a sphere, substitute its coordinates into the left-hand side and compare with r².
例如，球心在原点、半径为5的球面方程为x² + y² + z² = 25。要判断给定点是在球面上、球内还是球外，将该点坐标代入等式左侧，与r²进行比较。
9. Coordinates in the Context of Solids | 立体图形中的坐标
Many IB problems ask you to work with vertices of a cuboid or pyramid placed in a coordinate system. The key technique is to read off coordinates systematically from the diagram, using the given side lengths.
许多IB题目要求在坐标系中处理长方体或棱锥的顶点。关键技巧是根据图形和给定的边长，系统地读出各顶点坐标。
For instance, a cuboid with dimensions 4 × 3 × 5 placed at the origin has vertices at all eight combinations of 0 or 4 in x, 0 or 3 in y, and 0 or 5 in z. The coordinates of the center of the cuboid are simply (2, 1.5, 2.5).
例如，一个长4、宽3、高5且在原点处的长方体，其八个顶点的坐标是x取0或4、y取0或3、z取0或5的所有组合。该长方体的中心坐标就是(2, 1.5, 2.5)。
10. Distance from a Point to the Axes and Planes | 点到轴和平面的距离
You should be able to find the perpendicular distance from a point to each coordinate plane. For a point P(x, y, z):
你需要能够求出点到各个坐标平面的垂直距离。对于点P(x, y, z)：
| Plane / Axis | Distance |
| xy-plane (z = 0) | |z| |
| yz-plane (x = 0) | |x| |
| xz-plane (y = 0) | |y| |
| x-axis | √(y² + z²) |
Notice: the distance from a point to a plane is simply the absolute value of the coordinate that is not involved in that plane. The distance to an axis requires the Pythagorean theorem with the other two coordinates.
注意：点到平面的距离就是该平面未涉及的那个坐标的绝对值。点到轴的距离则需要用另外两个坐标通过勾股定理计算。
11. Common Pitfalls in 3D Coordinate Problems | 三维坐标问题的常见误区
Students often make avoidable errors in 3D coordinate questions. The most common pitfalls include confusing axes, forgetting to apply signs for negative coordinates, and misinterpreting three-figure bearings. Let’s examine these one by one.
学生在三维坐标问题中经常犯一些可以避免的错误。最常见的误区包括：混淆坐标轴、忘记负数坐标的符号、以及误解三维方位角。我们逐一分析。
- Mixing up x, y, and z: always write coordinates in the order (x, y, z) without skipping commas.
- Forgetting that negative coordinates place the point on the opposite side of the origin along that axis.
- Using the 2D distance formula in 3D problems — always include the z term.
- Not rationalising the final answer when required by the question.
- 混淆x、y和z：始终按(x, y, z)的顺序书写坐标，不要省略逗号。
- 忘记负坐标将点放置在对应轴的相反方向。
- 在三维问题中使用二维距离公式——始终包含z项。
- 题目要求时未对最终结果进行分母有理化。
12. Exam Strategy and Practice Tips | 考试策略与练习建议
To excel in questions about 3D coordinates, practice visualizing points before calculating. Draw a small 3D sketch whenever possible. This helps you catch sign errors and ensures you understand the spatial relationship between points.
要在三维点坐标问题中表现出色，请在计算之前练习可视化确定点的位置。尽可能画一个小型三维草图。这有助于你发现符号错误，并确保你理解点之间的空间关系。
When using a graphing calculator, remember to input coordinates in the correct order. In IB exams, show your working clearly: the method marks are often worth more than the final answer. For paper 2 questions, use the calculator efficiently but still write down the formula you are applying.
使用绘图计算器时，注意以正确顺序输入坐标。在IB考试中，清晰地展示你的解题过程：方法分通常比最终答案更值钱。对于卷二的问题，高效使用计算器，但仍要写出你所应用的公式。
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