📚 F - The vector between two points | F - 两点间的向量
Vectors are a cornerstone of the IB Mathematics curriculum, appearing in both Analysis and Approaches and Applications and Interpretation. One of the most fundamental skills is finding the vector that connects two points in two or three dimensions. Understanding how to form this vector correctly unlocks topics such as line equations, distances, and geometric proofs. This article will guide you through everything you need to know about the vector between two points, from position vectors and notation to magnitude, unit vectors, and common pitfalls.
向量是IB数学课程的核心内容之一，同时出现在分析与方法和应用与解释两个课程中。其中最基础的技能之一就是求连接平面或空间中两点的向量。正确掌握这一知识点有助于你进一步学习直线方程、距离以及几何证明等专题。本文将为你全面讲解两点间的向量，包括位置向量、记法、模长、单位向量以及常见错误。
1. Understanding Position Vectors | 理解位置向量
Every point in a coordinate plane or space can be represented by a position vector. A position vector starts at the origin O and ends at the given point. For a point A with coordinates (x₁, y₁) in 2D or (x₁, y₁, z₁) in 3D, its position vector is typically written in bold as a. We identify a with the column or row of its components: a sition vector simply tells us how to get from the origin to that point in a straight line.
坐标系中的每一个点都可以用一个位置向量来表示。位置向量从原点O出发，终止于该点。对于坐标为 (x₁, y₁) 的二维点A或坐标为 (x₁, y₁, z₁) 的三维点A，其位置向量通常用粗体 a 表示。我们直接把 a 和它的分量联系在一起：a = (x₁, y₁) 或 (x₁, y₁, z₁)。位置向量实际上就是告诉我们如何沿着直线从原点到达该点。
When we have two points A and B, their position vectors a and b are the building blocks for the vector that connects them. It is essential to remember that a and b are rooted at the origin, while the vector between the points will have a different starting point and direction.
当我们有两个点A和B时，它们的位置向量 a 和 b 就是构建连接两点向量的基本要素。务必牢记 a 和 b 都以原点为起点，而两点之间的向量则具有完全不同的起点和方向。
2. Defining the Vector Between Two Points | 定义两点间的向量
Suppose you have two points A and B with position vectors a = (x₁, y₁, z₁) and b = (x₂, y₂, z₂). The vector that takes you from A to B is written as AB and is defined by subtracting the coordinates of A from those of B. In vector terms, this is simply the difference of the two position vectors:
假设有两个点A和B，它们的位置向量分别为 a = (x₁, y₁, z₁) 和 b = (x₂, y₂, z₂)。从A指向B的向量记作 AB，由B的坐标减去A的坐标得到。用向量语言来说，它就是两个位置向量的差：
AB = b − a
This relationship is true regardless of whether you are working in two dimensions or three dimensions. In component form, it looks like this:
无论你在二维还是三维空间中运算，这个关系都成立。用分量形式表示如下：
AB = (x₂ − x₁, y₂ − y₁, z₂ − z₁)
The subtraction ensures that we start at point A, follow the reversed position vector −a back to the origin, and then follow b to point B. The net effect is exactly the journey from A to B.
这个减法确保了我们先从点A出发，沿着反向位置向量 −a 回到原点，然后再沿 b 到达点B。净效果正是从A到B的路径。
3. Vector Notation and Coordinate Systems | 向量记法与坐标系
In IB Mathematics, you will see vectors expressed in several equivalent notational forms. The most common are column vectors, row vectors, and unit vector combinations. For a vector v = (3, −2, 5), all of the following represent the same thing:
在IB数学中，你会看到向量以几种等价的形式出现。最常见的是列向量、行向量和单位向量的组合。对于一个向量 v = (3, −2, 5)，以下所有表示都是等价的：
- Row vector: (3, −2, 5)
- Column vector: being written vertically as 3 on top, −2 in the middle, 5 at the bottom (not easy to format here, but you will see it in your textbook).
- Using i, j, k: 3i − 2j + 5k
当我们需要写出从A到B的向量时，可以直接用坐标差给出。例如，若A(1, 4, −3)且B(6, 2, 0)，则 AB = (6−1, 2−4, 0−(−3)) = (5, −2, 3)。保持分量顺序一致是关键。
When we need to write the vector from A to B, we can directly give the coordinate differences. For instance, if A(1, 4, −3) and B(6, 2, 0), then AB = (6−1, 2−4, 0−(−3)) = (5, −2, 3). Keeping the order of components consistent is crucial.
4. Visualizing the Vector from A to B | 可视化从A到B的向量
A clear geometric picture helps avoid sign errors. Draw points A and B in a coordinate grid. The vector AB is the arrow that starts at A and points toward B. If you move the same arrow so that its tail sits at the origin, its head will land at the point with coordinates equal to the components of AB. This translated arrow is exactly the position vector b − a.
清晰的几何图像有助于避免符号错误。在坐标网格中标出点A和B。向量 AB 是一支从A出发指向B的箭头。如果你把这支箭头平移到原点，箭头的末端将落在坐标等于 AB 各分量的点上。平移后的箭头正好就是位置向量 b − a。
In 3D the idea is the same, though we imagine the difference in the z-coordinate as moving vertically. Understanding the geometric subtraction b − a as “head minus tail” reinforces the correct order of subtraction: B minus A, not the other way around.
在三维空间中思路相同，只是我们把z坐标差理解为垂直方向的移动。将几何减法 b − a 理解为“终点减起点”有助于强化正确的减法顺序：B减A，而不是反过来。
5. Calculating the Vector AB Step by Step | 逐步计算向量AB
Let's walk through a concrete example. Take two points in 3D: P(2, −1, 4) and Q(7, 3, −2). To find the vector PQ:
我们来看一个具体例子。取三维空间中的两个点：P(2, −1, 4) 和 Q(7, 3, −2)。计算向量 PQ：
Step 1: Write down the coordinates. p = (2, −1, 4), q = (7, 3, −2).
of P from Q: x-component: 7 − 2 = 5; y-component: 3 − (−1) = 4; z-component: −2 − 4 = −6.
Step 3: Combine: PQ = (5, 4, −6).
步骤1：写下坐标。p = (2, −1, 4), q = (7, 3, −2)。
步骤2：用Q的坐标减去P的坐标：x分量：7 − 2 = 5；y分量：3 − (−1) = 4；z分量：−2 − 4 = −6。
步骤3：组合起来：PQ = (5, 4, −6)。
at Q: (2+5, −1+4, 4+(−6)) = (7, 3, −2). This is an excellent habit to confirm your answer.
你可以验证一下：从P出发加上这个向量确实到达Q：(2+5, −1+4, 4+(−6)) = (7, 3, −2)。养成验证答案的好习惯非常有用。
6. The Negative Vector from B to A | 从B到A的负向量
If you reverse the order of subtraction, you obtain the vector that goes from B back to A, denoted BA. Clearly:
如果把减法顺序反过来，就得到了从B回到A的向量，记作 BA。显然：
BA = a − b = −(b − a) = −AB
So BA is simply the negative of AB. The vectors AB and BA have the same magnitude but opposite directions. This property is extremely useful when rearranging vector equations or proving geometric results.
因此 BA 就是 AB 的负向量。AB 和 BA 大小相同但方向相反。在整理向量方程或证明几何结论时，这个性质极为有用。
For example, if you are given AB = (3, −1, 2) and asked for BA, you can immediately write (−3, 1, −2) without recalculating. Many students slip up by mixing the direction; always double-check which point is the tail and which is the head.
例如，假设已知 AB = (3, −1, 2)，要求 BA，你可以立刻写出 (−3, 1, −2) 而无需重新计算。很多学生会因为混淆方向而犯错，所以一定要再次确认哪个点是起点、哪个点是终点。
7. Magnitude of the Vector Between Two Points | 两点间向量的模
The magnitude (or length) of the vector AB is the distance between points A and B. Using the standard distance formula derived from Pythagoras’ theorem, we have for 2D:
向量 AB 的模（或长度）就是点A和点B之间的距离。利用由勾股定理得出的距离公式，在二维空间中有：
|AB| = √[(x₂ − x₁)² + (y₂ − y₁)²]
And for 3D:
而在三维空间中：
|AB| = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
Continuing with the earlier example P(2, −1, 4) and Q(7, 3, −2), we have PQ = (5, 4, −6). Its magnitude is |PQ| = √(5² + 4² + (−6)²) = √25 + 16 + 36 = √77. This represents the straight-line distance from P to Q.
继续用前面的例子P(2, −1, 4)和Q(7, 3, −2)，我们有 PQ = (5, 4, −6)。它的模为 |PQ| = √(5² + 4² + (−6)²) = √25 + 16 + 36 = √77。这表示从P到Q的直线距离。
The magnitude is always a non-negative scalar. It is crucial in calculating unit vectors and in many kinematics and force problems where only the size of a vector matters.
模长永远是一个非负的标量。在计算单位向量以及许多只需要向量大小的运动学和力学问题中，它至关重要。
8. Unit Vector in the Direction of AB | 在AB方向上的单位向量
A unit vector has a magnitude of 1 and points in a specific direction. To obtain the unit vector in the direction of AB, divide AB by its own magnitude:
单位向量的模为1，并指向某一特定方向。要得到沿 AB 方向的单位向量，只需用 AB 除以其自身的模：
û = AB / |AB|
For PQ = (5, 4, −6) and |PQ| = √77, the unit vector is (5/√77, 4/√77, −6/√77). You can leave it in this exact form or simplify if possible, but rationalizing the denominator is generally not required in IB unless specified.
对于 PQ = (5, 4, −6) 和 |PQ| = √77，单位向量为 (5/√77, 4/√77, −6/√77)。你可以保留这种精确形式，或者尽可能化简，但IB考试中通常不要求分母有理化，除非题目明确说明。
Unit vectors are essential when you need to describe a direction independently of distance. They appear in topics such as vector equations of lines and in finding the angle between two vectors using dot products.
当需要抛开距离单独描述方向时，单位向量就变得必不可少。它们在直线向量方程和运用点积求两向量夹角等专题中都会出现。
9. Midpoint of Two Points Using Vectors | 利用向量求两点中点
Another elegant application of position vectors is the midpoint formula. If M is the midpoint of segment AB, its position vector m is the average of a and b:
位置向量的另一个巧妙应用是中点公式。若M是线段AB的中点，其位置向量 m 是 a 和 b 的平均值：
m = (a + b) / 2
((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). So if we know the vector between two points, we can also easily describe the location of their midpoint.
用坐标表示，即 M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)。因此，如果我们知道两点间的向量，就能方便地描述它们中点的位置。
This relationship comes directly from vector addition: starting at A, going half of the vector AB takes you to M. Thus m = a + ½ AB = a + ½(b − a) = (a+b)/2. The midpoint formula appears often in parallelogram and quadrilateral proofs.
这一关系直接来自向量加法：从A出发，沿 AB 走一半就到达M。因此 m = a + ½ AB = a + ½(b − a) = (a+b)/2。中点公式在平行四边形和四边形证明中经常出现。
10. Applications in Geometry and Physics | 在几何与物理中的应用
The vector between two points is more than a computational exercise; it is the foundation of many geometric and physical problems. In geometry, it helps prove collinearity, determine whether points form a triangle, or find the fourth vertex of a parallelogram. For instance, if AB is parallel to CD, then AB = k CD for some scalar k.
两点间的向量不仅仅是一道计算题，它是许多几何与物理问题的基础。在几何中，它可用来证明共线性、判断是否构成三角形或求出平行四边形的第四个顶点。例如，若 AB 与 CD 平行，则存在某个标量k使得 AB = k CD。
In physics, the displacement vector of a particle that moves from point r₁ to r₂ is exactly r₂ − r₁. This displacement is then used to compute velocity, acceleration, and work done by a force. The ability to find the vector between two points is thus implicitly used whenever positions change over time.
在物理中，质点从位置r₁移动到r₂的位移向量正好就是 r₂ − r₁。这一位移进而被用于计算速度、加速度和力做的功。因此，只要位置随时间变化，求两点间向量的技能就在无形中被应用。
11. Worked Examples | 例题精解
Example 1: Given A(−2, 5) and B(3, −1), find AB, its magnitude, and the unit vector in its direction.
例1： 已知 A(−2, 5) 和 B(3, −1)，求 AB、其模以及方向上的单位向量。
Solution: AB = (3 − (−2), −1 − 5) = (5, −6). Magnitude |AB| = √(5² + (−6)²) = √25+36 = √61. Unit vector û = (5/√61, −6/√61).
解答： AB = (3 − (−2), −1 − 5) = (5, −6)。模长 |AB|
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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