Vectors and Vector Notation | 向量与向量表示法

📚 Vectors and Vector Notation | 向量与向量表示法

Vectors form a fundamental part of IGCSE Mathematics, appearing in coordinate geometry, transformations and geometric proofs. A firm grasp of vector notation and the rules for combining vectors is essential for tackling topics like translations, vector magnitude and problems involving parallel or collinear segments. This guide walks through every key idea, from basic representation to geometric reasoning, using clear notation and step‑by‑step examples tailored to the Edexcel International GCSE specification.

向量是 IGCSE 数学的基础内容，广泛出现在坐标几何、变换与几何证明中。牢固掌握向量符号及其运算法则，对于解决平移、向量模长以及平行或共线线段等问题至关重要。本指南将逐项解析每一个关键概念，从基本表示法到几何推理，采用清晰的符号和循序渐进的范例，紧贴 Edexcel International GCSE 考纲。

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1. Introduction to Vectors | 向量的引入

A scalar quantity has only magnitude (size). Examples include mass (5 kg), speed (10 m/s), distance (8 km) and time (3 s). A vector quantity has both magnitude and direction. Typical vectors are displacement (5 km east), velocity (10 m/s upwards), force (50 N to the left) and acceleration. In IGCSE problems, vectors are most often used to describe translations and to work with geometric relationships in the coordinate plane.

标量只有大小（模），例如质量（5 kg）、速率（10 m/s）、距离（8 km）和时间（3 s）。向量既有大小又有方向，常见的向量包括位移（向东 5 km）、速度（向上 10 m/s）、力（向左 50 N）和加速度。在 IGCSE 试题中，向量最常用于描述平移以及在坐标平面中处理几何关系。

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2. Vector Notation: Column Vectors and Component Form | 向量符号：列向量与分量形式

Vectors are written either as a bold letter, such as a or v, or as a letter with an underline in handwriting, e.g. a. In IGCSE, a two‑dimensional vector is often expressed as a column vector like

(	3	)
	4

, which means ‘3 units in the positive x‑direction and 4 units in the positive y‑direction’. This can also be written in component form as v , though the column vector style is very common in Edexcel papers. The top number is the x‑component and the bottom number is the y‑component. A general vector can be written as

(	x	)
	y

or (x, y).

向量可用粗体字母表示，如 a 或 v；手写时则在字母下方加下划线，如 a。在 IGCSE 中，二维向量常写为列向量的形式，例如

(	3	)
	4

，表示「沿 x 轴正方向 3 个单位，沿 y 轴正方向 4 个单位」。它也可以写成分量形式 v = (3, 4)，不过 Edexcel 试卷中列向量更为常见。上方数字为 x 分量，下方数字为 y 分量。一般向量可记为

(	x	)
	y

或 (x, y)。

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3. Geometric Representation of Vectors | 向量的几何表示

A vector can be drawn as a directed line segment. The length of the segment represents the magnitude, while the arrowhead indicates the direction. For example, AB starts at point A and ends at point B. In coordinate geometry, if A(1, 2) and B(4, 6), the vector AB is found by subtracting the coordinates:

(	4 – 1	)
	6 – 2

=

(	3	)
	4

. This ‘end‑minus‑start’ rule is used repeatedly in vector geometry.

向量可以画成一条有向线段。线段的长度表示向量的模，箭头则指明方向。例如，AB 从点 A 出发，终点为点 B。在坐标几何中，若 A(1, 2)、B(4, 6)，则向量 AB 可通过坐标相减求得：

(	4 – 1	)
	6 – 2

=

(	3	)
	4

。这种「终点减起点」的规则在向量几何中反复使用。

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4. Equality of Vectors | 向量相等

Two vectors are equal if they have the same magnitude and the same direction. Their position in the plane does not matter; a vector can be translated (moved) without changing its value. For instance,

(	5	)
	-2

and

(	5	)
	-2

are equal wherever they are drawn. When working with directed segments, AB = CD if and only if the quadrilateral ABDC is a parallelogram (note the order of vertices).

若两个向量的大小相等且方向相同，则二者相等。它们在平面上的位置无关紧要——向量可以平移，其值不变。例如，

(	5	)
	-2

与

(	5	)
	-2

无论画在哪里都相等。在处理有向线段时，AB = CD 当且仅当四边形 ABDC 是平行四边形（注意顶点的顺序）。

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5. Negative Vectors | 负向量

The negative of a vector a is written as –a. It has the same magnitude as a but points in exactly the opposite direction. In column form, if a =

(	x	)
	y

, then –a =

(	-x	)
	-y

. Likewise, BA = –AB, a fact very useful when moving backwards along a segment.

向量 a 的负向量记作 –a。它与 a 大小相等，但方向完全相反。在列向量形式中，若 a =

(	x	)
	y

，则 –a =

(	-x	)
	-y

。同理，BA = –AB，这个结论在沿线段逆向移动时非常实用。

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6. Scalar Multiplication of Vectors | 向量的标量乘法

When a vector is multiplied by a scalar (a real number) k, its magnitude is scaled by |k|. If k > 0, the direction stays the same; if k < 0, the direction is reversed. In column form, k

(	x	)
	y

=

(	kx	)
	ky

. For example, if p =

(	2	)
	-3

, then 3p =

(	6	)
	-9

and –2p =

(	-4	)
	6

. Scalar multiplication is frequently used to create parallel vectors and to prove collinearity.

向量乘以一个标量（实数）k 时，其模会缩放 |k| 倍。若 k > 0，方向不变；若 k < 0，方向反向。列向量形式下， k

(	x	)
	y

=

(	kx	)
	ky

。例如，若 p =

(	2	)
	-3

，则 3p =

(	6	)
	-9

，–2p =

(	-4	)
	6

。标量乘法常用于构造平行向量以及证明共线。

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7. Addition and Subtraction of Vectors | 向量的加减法

To add two vectors given in column form, simply add the corresponding components:

(	a	)
	b

+

(	c	)
	d

=

(	a + c	)
	b + d

. Subtraction works similarly:

(	a	)
	b

–

(	c	)
	d

=

(	a – c	)
	b – d

. For example, if u = (2, 5) and v = (-1, 3), then u + v = (1, 8) and u – v = (3, 2). Geometrically, subtracting v is the same as adding –v.

两个以列向量形式给出的向量相加时，只需将对应分量相加：

(	a	)
	b

+

(	c	)
	d

=

(	a + c	)
	b + d

。减法类似：

(	a	)
	b

–

(	c	)
	d

=

(	a – c	)
	b – d

。例如，若 u = (2, 5)，v = (-1, 3)，则 u + v = (1, 8)，u – v = (3, 2)。从几何上看，减 v 等同于加上 –v。

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8. The Triangle and Parallelogram Laws | 三角形法则与平行四边形法则

The triangle law states that for any points A, B and C, AB + BC = AC. To add p and q geometrically, place the tail of q at the head of p; the sum is the vector from the tail of p to the head of q. The parallelogram law puts the tails of both vectors at the same point; the sum is the diagonal of the completed parallelogram. These laws are especially helpful when dealing with vector paths in shapes and when proving that one vector is the multiple of another.

三角形法则指出，对于任意三点 A、B、C，有 AB + BC = AC。要想用几何方式将 p 与 q 相加，可把 q 的尾端放到 p 的尖端；和向量就是从 p 的尾端指向 q 的尖端的向量。平行四边形法则则是将两个向量的尾端放在同一点上；和向量为所构成的平行四边形的对角线。这些法则在处理图形的向量路径以及证明一个向量是另一向量的倍数时格外有用。

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9. Position Vectors | 位置向量

A position vector starts at the origin O and ends at a point P. It is usually written as OP or simply as p when the origin is understood. If P has coordinates (x, y), then the position vector of P is

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(	x	)
	y