Vectors in Two Dimensions | 二维向量

📚 Vectors in Two Dimensions | 二维向量

Vectors are one of the most visual and rewarding topics in the IGCSE Edexcel Mathematics syllabus. They allow us to describe movement, direction and position algebraically, and they appear in many exam questions that combine geometry with algebra. This guide covers everything you need: notation, operations, magnitude, parallel vectors, position vectors and the classic ratio problems that carry the highest marks.

向量是 IGCSE Edexcel 数学大纲中最具图形化、回报也最高的主题之一。它让我们能用代数描述运动、方向和位置，并且经常出现在几何与代数结合的考题中。本指南涵盖你所需的一切：记号、运算、模（长度）、平行向量、位置向量，以及分值最高的经典比例分点问题。

1. What Is a Vector? | 什么是向量？

A vector is a quantity that has both magnitude (size) and direction. In contrast, a scalar has only magnitude. For example, “5 km north” is a vector, while “5 km” alone is a scalar. Displacement, velocity and force are vectors; distance, speed and mass are scalars.

向量是既有大小（模）又有方向的量。相比之下，标量只有大小。例如，“向北 5 公里”是向量，而单独的“5 公里”是标量。位移、速度和力是向量；距离、速率和质量是标量。

In two dimensions, a vector can be represented by a column vector, a bold letter such as a, or an arrow notation such as AB→. The vector from point A to point B is written AB→, while BA→ is exactly the opposite direction.

在二维平面中，向量可以用列向量、粗体字母（如 a）或箭头记号（如 AB→）表示。从点 A 指向点 B 的向量写作 AB→，而 BA→ 则方向完全相反。

2. Column Vectors and Notation | 列向量与记号

In IGCSE Edexcel, a vector is often written as a column vector. For example, the vector ⎛ 3 ⎞
⎝ 2 ⎠ means “3 units to the right and 2 units up”. The top number is the x-component; the bottom number is the y-component.

在 IGCSE Edexcel 中，向量通常写成列向量。例如，向量 ⎛ 3 ⎞
⎝ 2 ⎠ 表示“向右 3 个单位，向上 2 个单位”。上面的数是 x 分量，下面的数是 y 分量。

If a vector points left or down, its component is negative. The column vector ⎛ −2 ⎞
⎝ 5 ⎠ means 2 units left and 5 units up. Always keep the sign attached to the correct component.

如果向量指向左方或下方，其分量为负数。列向量 ⎛ −2 ⎞
⎝ 5 ⎠ 表示向左 2 个单位、向上 5 个单位。务必正确区分正负号所对应的分量。

3. Adding and Subtracting Vectors | 向量的加法与减法

To add vectors, add the corresponding components separately. Geometrically, place the tip of one vector at the tail of the other; the resultant vector goes from the first tail to the final tip. This is called the triangle law of addition.

向量相加时，将对应的分量分别相加。几何上，把一个向量的箭头接在另一个向量的箭尾上，合向量从第一个箭尾指向最后的箭头。这就是向量加法的三角形法则。

⎛ a ⎞ + ⎛ c ⎞ = ⎛ a + c ⎞
⎝ b ⎠ ⎝ d ⎠ ⎝ b + d ⎠

Subtraction is also done component by component. The key exam fact is that AB→ = OB→ − OA→. The order matters: AB→ is displacement from A to B, so you subtract the position vector of A from the position vector of B.

减法同样按分量进行。考试中最关键的事实是 AB→ = OB→ − OA→。顺序至关重要：AB→ 是从 A 到 B 的位移，因此要用 B 的位置向量减去 A 的位置向量。

4. Scalar Multiplication | 数与向量的乘法（缩放）

Multiplying a vector by a scalar k stretches its length by a factor of k. If k is negative, the direction reverses. Every component is multiplied by k separately.

向量乘以标量 k，其长度会变为原来的 k 倍。如果 k 为负数，则方向反转。将每个分量分别乘以 k 即可。

k × ⎛ x ⎞ = ⎛ kx ⎞
⎝ y ⎠ ⎝ ky ⎠

For example, 2 × ⎛ 3 ⎞
⎝ 2 ⎠ = ⎛ 6 ⎞
⎝ 4 ⎠. The vector keeps the same direction but doubles in length, whereas multiplying by −1 gives a vector of equal length pointing the opposite way.

例如，2 × ⎛ 3 ⎞
⎝ 2 ⎠ = ⎛ 6 ⎞
⎝ 4 ⎠。向量方向不变但长度变为两倍；而乘以 −1 则得到长度相等、方向相反的向量。

5. Magnitude of a Vector | 向量的模（长度）

The magnitude of a vector is its length, and it is found using Pythagoras’ theorem. For a column vector ⎛ x ⎞
⎝ y ⎠, the magnitude is written |a| or |AB→| and is calculated as √(x² + y²).

向量的模就是它的长度，用勾股定理计算。对于列向量 ⎛ x ⎞
⎝ y ⎠，其模记作 |a| 或 |AB→|，计算公式为 √(x² + y²)。

|a| = √(x² + y²)

Note that the magnitude is always a positive scalar, and it is measured in the same units as the original quantities. A unit vector has a magnitude of exactly 1.

注意，模总是正的标量，并且与原来的量使用相同单位。单位向量的模正好等于 1。

6. Parallel Vectors | 平行向量

Two non-zero vectors are parallel if one is a scalar multiple of the other. In other words, a is parallel to b if a = k b for some constant k, where k can be positive, negative or a fraction.

两个非零向量平行，当且仅当一个向量是另一个向量的数倍。也就是说，若存在常数 k，使得 a = k b，则 a 与 b 平行。k 可以是正数、负数或分数。

In exam questions, you are often asked to “show that PQ is parallel to AB”. This means you must demonstrate that the vector PQ→ equals a scalar multiple of AB→. It is not enough to say they look parallel; you must show the component-wise relationship.

考试中常要求“证明 PQ 平行于 AB”。这意味着你必须说明向量 PQ→ 等于 AB→ 的某个数倍。仅凭观察“看起来平行”是不够的，必须写出分量之间的关系。

PQ→ = k · AB→

7. Position Vectors | 位置向量

A position vector describes the location of a point relative to the origin O. If point A has coordinates (3, 4), then its position vector is OA→ = ⎛ 3 ⎞
⎝ 4 ⎠. Every point in the plane has a unique position vector.

位置向量描述一个点相对于原点 O 的位置。若点 A 的坐标为 (3, 4)，则其位置向量为 OA→ = ⎛ 3 ⎞
⎝ 4 ⎠。平面内每个点都有唯一的位置向量。

Position vectors are especially useful because they let us connect algebra with geometry. For any two points A and B with position vectors a and b, the displacement vector is AB→ = b − a. This beautifully expresses “how to get from A to B”.

位置向量特别有用，因为它将代数与几何联系起来。对于任意两点 A 和 B，若其位置向量分别为 a 和 b，则位移向量为 AB→ = b − a。这简洁地表达出“如何从 A 走到 B”。

8. Vector Geometry Problems | 向量与平面几何题

In vector geometry, the standard strategy is to express every vector in terms of two independent base vectors, usually labelled a and b. You then travel along routes around the diagram, following sides and diagonals, to write any requested vector.

在向量几何题中，常用策略是把所有向量用两个独立的基向量（通常记为 a 和 b）表示。然后沿着图形中的边和对角线“行走”，写出任意要求的向量。

Consider a triangle with vertices O, A and B, where OA→ = a and OB→ = b. If M is the midpoint of AB, then AM→ = ½ (b − a), so OM→ = a + ½ (b − a) = ½ a + ½ b = ½ (a + b).

考虑三角形 OAB，其中 OA→ = a，OB→ = b。若 M 是 AB 的中点，则 AM→ = ½ (b − a)，因此 OM→ = a + ½ (b − a) = ½ a + ½ b = ½ (a + b)。

OM→ = (a + b) / 2

9. Ratio Problems on a Straight Line | 直线上按比例分点

One of the highest-scoring questions in the IGCSE paper is the ratio division problem. If point P divides the line segment AB in the ratio m : n, meaning AP : PB = m : n, then P divides the line such that AP is m parts and PB is n parts of the whole.

IGCSE 考试中分值最高的题型之一是按比例分点问题。若点 P 将线段 AB 分为 m : n，即 AP : PB = m : n，那么 AP 占全长的 m 份，PB 占全长的 n 份。

Using the general formula with position vectors a for A and b for B, the position vector of P is:

利用 A 的位置向量 a 和 B 的位置向量 b，P 的位置向量的一般公式为：

OP→ = (n a + m b) / (m + n)

Worked example: A has position vector a, B has position vector b, and P lies on AB such that AP : PB = 2 : 1. Then OP→ = a + (2/3)(b − a) = (1/3)a + (2/3)b = (a + 2b) / 3.

例题：A 的位置向量为 a，B 的位置向量为 b，P 在 AB 上且 AP : PB = 2 : 1。则 OP→ = a + (2/3)(b − a) = (1/3)a + (2/3)b = (a + 2b) / 3。

10. Vector Proofs: Collinearity | 向量证明：三点共线

To prove that three points A, B and C are collinear (lie on the same straight line), you must show two things: that AB→ is a scalar multiple of BC→ (so the vectors are parallel), and that the two vectors share the common point B.

要证明三点 A、B、C 共线（在同一条直线上），必须证明两点：AB→ 是 BC→ 的数倍（即两向量平行），并且这两个向量有一个公共点 B。

Also, to prove that X is the midpoint of AB, show that AX→ = ½ AB→. To prove that a quadrilateral is a parallelogram, show that one pair of opposite sides is both equal and parallel, i.e. AB→ = DC→.

同理，要证明 X 是 AB 的中点，只需证明 AX→ = ½ AB→。要证明四边形是平行四边形，只需证明一组对边相等且平行，即 AB→ = DC→。

AB→ = k · BC→ ⟹ A, B, C are collinear

AB→ = k · BC→ ⟹ A、B、C 三点共线

11. Common Exam Pitfalls | 常见失分陷阱

Forgetting the subtraction order: AB→ = OB→ − OA→, not OA→ − OB→. The direction matters! | 忘记减法顺序：AB→ = OB→ − OA→，而不是 OA→ − OB→。方向至关重要！
Mixing x and y components when adding or subtracting vectors. Always line up the components vertically. | 加减向量时混淆 x 与 y 分量。务必按垂直方向对齐各分量。
Writing magnitude as negative: √x² of a squared term is always positive, so |a| ≥ 0. | 把模写成负数：平方项开根号总是非负，因此 |a| ≥ 0。
Forgetting to state the value of k in a parallel-vector proof. The marker needs to see the scalar explicitly. | 在平行向量证明中忘记写出 k 的值。阅卷者需要看到明确的数倍关系。
Using column vector brackets incorrectly, for example swapping the x and y components. | 列向量括号使用错误，比如把 x 和 y 分量写反。

12. Exam Strategy and Summary | 应考策略与要点总结

Vectors are a guaranteed part of the IGCSE Edexcel Mathematics exam. Start by writing down the two base vectors a and b, then express the target vector as a route through the diagram. Always show every step of your algebra, because method marks are awarded even if your final simplification is wrong.

向量是 IGCSE Edexcel 数学考试的必考内容。解题时先写出两个基向量 a 和 b，然后将目标向量表示为经过图形的某条路径。务必写出每一步代数运算，因为即使最后化简出错，过程分仍然可以获得。

Finally, check whether your final vector expression is fully simplified, and whether it factors neatly. In ratio questions, your answer should usually take the form (n a + m b) / (m + n), which is easy to verify by testing endpoint values. With regular practice, vectors move from “hardest topic” to “guaranteed marks”.

最后，检查最终向量表达式是否已经彻底化简、是否可以提取公因式。在比例分点题中，答案通常应写成 (n a + m b) / (m + n) 的形式，也可以通过代入端点值来检验。只要勤加练习，向量就会从“最难的模块”变成“稳拿分的题型”。

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Vectors in Two Dimensions | 二维向量