IGCSE Mathematics Scalar Multiplication of Vectors — IGCSE数学：向量的数乘运算

一、什么是向量：位移背后的数学语言 | What Is a Vector? The Mathematical Language of Displacement

在 IGCSE 数学中，我们把量分为两大类：标量（scalar）和向量（vector）。标量只有大小（magnitude），没有方向，比如温度、质量、时间和路程；向量既有大小又有方向，比如位移、速度和力。举例来说，说”这辆车开了 50 公里”是一个标量描述，因为只有距离；而说”这辆车从上海向东开了 50 公里”就是一个向量描述，因为既有距离又有方向。

In IGCSE Mathematics, quantities are divided into two broad classes: scalars and vectors. A scalar has magnitude only, with no direction – examples include temperature, mass, time and distance. A vector has both magnitude and direction – examples include displacement, velocity and force. For instance, saying “the car travelled 50 km” is a scalar description because it gives distance only, while “the car travelled 50 km east from Shanghai” is a vector description because it gives both distance and direction.

向量在生活中的应用非常广泛：导航系统用向量计算航向和距离，物理学家用向量分析力的合成，游戏引擎用向量描述角色的移动。在 IGCSE 考试中，向量是 Edexcel 考纲的必考内容，通常出现在试卷的后半部分，与几何证明、比例和坐标系结合考查。掌握向量的数乘运算，是理解整个向量章节的基石。

Vectors are used widely in real life: navigation systems use vectors to compute headings and distances, physicists use vectors to analyse combined forces, and game engines use vectors to describe character movement. In the IGCSE examination, vectors are a compulsory part of the Edexcel specification and usually appear in the later sections of the paper, combined with geometry proofs, ratios and coordinate systems. Mastering scalar multiplication of vectors is the foundation of the whole vectors chapter.

二、向量的表示方法：列向量与坐标分量 | Representing Vectors: Column Notation and Components

在 IGCSE Edexcel 课程中，向量最常见的表示方法是列向量（column vector）。一个列向量写成上下排列的两个数字，例如向量 a 可以写成 (4, -2)，其中上面的数字 4 表示水平方向的分量（向右为正），下面的数字 -2 表示垂直方向的分量（向上为正）。这种写法本质上和平面直角坐标系中的坐标一致：向量 (4, -2) 可以理解为”向右移动 4 个单位，再向下移动 2 个单位”。

In the IGCSE Edexcel course, the most common way to represent a vector is the column vector. A column vector is written as two numbers arranged one above the other. For example, vector a can be written as (4, -2), where the top number 4 is the horizontal component (positive to the right) and the bottom number -2 is the vertical component (positive upwards). This notation is essentially the same as a coordinate in the Cartesian plane: the vector (4, -2) can be read as “move 4 units right, then 2 units down”.

例如，从点 A(1, 3) 到点 B(5, 1) 的位移向量就是 AB = (5 – 1, 1 – 3) = (4, -2)。注意：向量 AB 表示从 A 出发到达 B 的位移，箭头从 A 指向 B。如果反过来写 BA，则 BA = (-4, 2)，方向完全相反。两个向量相等，当且仅当它们的对应分量分别相等；一个向量的负向量，就是把两个分量都取相反数。

For example, the displacement vector from point A(1, 3) to point B(5, 1) is AB = (5 – 1, 1 – 3) = (4, -2). Note that vector AB represents the displacement starting at A and arriving at B, with the arrow pointing from A to B. Written the other way round, BA = (-4, 2), which points in exactly the opposite direction. Two vectors are equal if and only if their corresponding components are equal; the negative of a vector is obtained by taking the opposite sign of both components.

在书写列向量时有一个经典易错点：不要把水平分量和垂直分量的顺序写反。水平分量永远写在上面。判断方法是联想坐标系：横坐标 x 在前，纵坐标 y 在后，列向量里 x 同样放在上方。考试中很多同学因为把 (4, -2) 写成 (-2, 4) 而丢掉整道题的分数，这是完全可以避免的失误。

There is a classic pitfall when writing column vectors: do not swap the order of the horizontal and vertical components. The horizontal component always goes on top. A useful memory aid is the coordinate system: x comes before y, and in a column vector x is likewise placed on top. In exams, many students lose the marks of an entire question because they write (-2, 4) instead of (4, -2) – a mistake that is entirely avoidable.

三、数乘的定义：用标量缩放向量 | The Definition of Scalar Multiplication: Scaling a Vector by a Number

数乘（scalar multiplication）就是把一个向量乘以一个数（这个数在数学上称为标量）。规则非常简单：把向量的每一个分量都乘以这个数。如果向量 a = (x, y)，那么 ka = (kx, ky)。例如，若 a = (3, -1)，则 2a = (6, -2)，5a = (15, -5)，(-2)a = (-6, 2)。注意每个分量都必须乘以 k，只乘其中一个分量是错误的。

Scalar multiplication means multiplying a vector by a number (called a scalar in mathematics). The rule is very simple: multiply every component of the vector by that number. If vector a = (x, y), then ka = (kx, ky). For example, if a = (3, -1), then 2a = (6, -2), 5a = (15, -5) and (-2)a = (-6, 2). Note that every component must be multiplied by k – multiplying only one component is a mistake.

数乘的运算性质与普通代数非常相似：结合律 k(ma) = (km)a，分配律 (k + m)a = ka + ma，以及 k(a + b) = ka + kb。这些性质说明，数乘和向量的加减法可以像代数式一样自由化简。1a = a，(-1)a = –a，0a = 0（零向量）。零向量是所有分量都为 0 的向量，它是向量加法的”零元素”。

The algebraic properties of scalar multiplication are very similar to ordinary algebra: associativity k(ma) = (km)a, distributivity (k + m)a = ka + ma, and k(a + b) = ka + kb. These properties mean that scalar multiplication and vector addition/subtraction can be simplified freely like algebraic expressions. We also have 1a = a, (-1)a = –a and 0a = 0 (the zero vector). The zero vector has every component equal to 0, and it acts as the “zero element” for vector addition.

四、数乘的几何意义：伸缩、反向与零向量 | The Geometric Meaning: Stretching, Reversing and the Zero Vector

数乘的几何意义非常直观：把向量 a 变成 ka，相当于把原来的箭头按比例缩放。当 k 大于 1 时，向量变长，方向不变；当 k 在 0 和 1 之间时，向量变短，方向不变；当 k 是负数时，向量不仅缩放，方向还会反转 180 度。例如，a = (2, 1) 指向右上方，2a = (4, 2) 仍然指向右上方但长度是原来的两倍，而 –a = (-2, -1) 指向左下方，长度不变。

The geometric meaning of scalar multiplication is very intuitive: turning vector a into ka means scaling the original arrow by a factor. When k is greater than 1, the vector becomes longer and keeps its direction; when k lies between 0 and 1, the vector becomes shorter and keeps its direction; when k is negative, the vector is scaled and also reversed through 180 degrees. For example, a = (2, 1) points up and to the right; 2a = (4, 2) still points up and to the right but is twice as long; –a = (-2, -1) points down and to the left with the same length.

理解”方向不变”的准确含义很重要：两个非零向量 ka 和 a（k 不等于 0）总是位于同一条直线上，我们称它们平行。当 k 大于 0 时方向相同（同向平行），当 k 小于 0 时方向相反（反向平行）。无论 k 取什么值，缩放后的向量都与原向量共线。这一性质是后面判断平行向量和共线点的理论基础。

It is important to understand the precise meaning of “direction unchanged”: two non-zero vectors ka and a (with k not equal to 0) always lie on the same straight line, and we say they are parallel. When k is positive they have the same direction (parallel in the same sense); when k is negative they have opposite directions (parallel in opposite senses). Whatever value k takes, the scaled vector is collinear with the original vector. This property is the theoretical basis for identifying parallel vectors and collinear points later.

还有一个特殊情形：当 k = 0 时，ka = 0，得到零向量。零向量的方向没有定义，长度为零。在考试中，如果题目问”向量 a 与向量 b 平行”，并且允许其中一个为零向量，答案会变得平凡，所以 IGCSE 题目通常约定所讨论的向量都是非零向量。做题时注意这个隐含条件。

There is one special case: when k = 0, ka = 0, giving the zero vector. The zero vector has undefined direction and zero length. In exams, if a question asks whether vector a is parallel to vector b, and one of them is allowed to be the zero vector, the answer becomes trivial – so IGCSE questions normally assume the vectors involved are non-zero. Keep this implicit condition in mind when solving problems.

五、平行向量的判定：数乘检验法 | Testing for Parallel Vectors: The Scalar Multiple Test

数乘最重要的应用之一就是判定两个向量是否平行。两个非零向量 a 和 b 平行，当且仅当存在一个非零实数 k，使得 b = ka。换句话说，如果一个向量的两个分量分别都是另一个向量对应分量的同一个倍数，那么这两个向量平行。例如，a = (2, 5)，b = (6, 15)，因为 6 = 3 × 2 且 15 = 3 × 5，所以 b = 3a，二者平行。

One of the most important applications of scalar multiplication is testing whether two vectors are parallel. Two non-zero vectors a and b are parallel if and only if there exists a non-zero real number k such that b = ka. In other words, if each component of one vector is the same multiple of the corresponding component of the other, the two vectors are parallel. For example, a = (2, 5) and b = (6, 15): since 6 = 3 x 2 and 15 = 3 x 5, we have b = 3a, so they are parallel.

检验的方法是”交叉比较”：先计算第一个分量的比值 k1 = bx / ax，再计算第二个分量的比值 k2 = by / ay。如果 k1 = k2，则平行；如果两个比值不相等，则不平行。例如 p = (4, 6) 与 q = (6, 10)：k1 = 6/4 = 1.5，k2 = 10/6 约等于 1.667，两个比值不同，所以 p 与 q 不平行。注意：当分母含有负号时，比值也要带上符号，负号不能丢失。

The test method is “cross comparison”: first compute the ratio of the first components k1 = bx / ax, then the ratio of the second components k2 = by / ay. If k1 = k2, they are parallel; if the two ratios differ, they are not. For example, p = (4, 6) and q = (6, 10): k1 = 6/4 = 1.5 while k2 = 10/6 is approximately 1.667; the ratios differ, so p and q are not parallel. Note that when a denominator is negative, the ratio must keep the negative sign – do not drop it.

平行的概念还可以推广到三个点共线：如果三点 A、B、C 满足向量 AB = k 乘以向量 AC（或 BC 与 AB 成比例），那么 A、B、C 三点共线。这是因为 AB 和 AC 共起点 A，它们平行又共点，只能落在同一条直线上。这种”向量成比例证明共线”的方法在 Edexcel IGCSE 的几何证明大题中几乎每年都会出现。

The concept of parallelism extends to collinearity of three points: if points A, B and C satisfy vector AB = k times vector AC (or BC is proportional to AB), then A, B and C are collinear. This is because AB and AC share the starting point A; being parallel and sharing a point, they must lie on the same straight line. This “proportional vectors prove collinearity” method appears in the Edexcel IGCSE geometry proof questions almost every year.

六、数乘与加减法的结合：化简向量表达式 | Combining Scalar Multiplication with Addition and Subtraction

在考试中，向量题常常要求你把形如 3a + 2b – a + 4b 的表达式化简。化简的规则与代数完全相同：先做数乘，再把同类的向量合并。这里”同类”指的是同一个向量的倍数。例如，3a + 2b – a + 4b = (3a – a) + (2b + 4b) = 2a + 6b。

In exams, vector questions often ask you to simplify expressions such as 3a + 2b – a + 4b. The simplification rules are exactly the same as in algebra: perform the scalar multiplication first, then combine like vectors. Here “like” means multiples of the same vector. For example, 3a + 2b – a + 4b = (3a – a) + (2b + 4b) = 2a + 6b.

如果给定了具体分量，例如 a = (2, -1)，b = (0, 3)，那么可以代入计算：3a + 2b = 3(2, -1) + 2(0, 3) = (6, -3) + (0, 6) = (6, 3)。代入时注意两个要点：第一，每个向量都要完整地套上括号再乘；第二，加法是对应分量相加，即 (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)。

If specific components are given, for example a = (2, -1) and b = (0, 3), you can substitute and compute: 3a + 2b = 3(2, -1) + 2(0, 3) = (6, -3) + (0, 6) = (6, 3). Two points to note when substituting: first, bracket each vector completely before multiplying; second, addition adds corresponding components, that is (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2).

减法可以理解为加上负向量：a – b = a + (-b)。而 –b 正是数乘 (-1)b，所以 a – b = (x1 – x2, y1 – y2)。这与”终点减起点”的口诀一致：从 A 到 B 的向量 AB = b – a（其中 a、b 分别是 A、B 的位置向量），即”后到的点减去先到的点”。

Subtraction can be understood as adding the negative vector: a – b = a + (-b). Since –b is precisely the scalar product (-1)b, we get a – b = (x1 – x2, y1 – y2). This agrees with the well-known rule “end point minus start point”: the vector from A to B is AB = b – a (where a and b are the position vectors of A and B), that is, “the later point minus the earlier point”.

七、单位向量：用数乘构造长度为 1 的向量 | Unit Vectors: Using Scalar Multiplication to Build Vectors of Length 1

向量的长度（模）用两个竖线表示，记作 |a|。如果 a = (x, y)，那么它的模为 |a| = sqrt(x^2 + y^2)，这正是勾股定理在坐标系中的体现：水平分量和垂直分量构成直角三角形的两条直角边，向量本身是斜边。例如 a = (3, 4)，则 |a| = sqrt(9 + 16) = 5。

The length (magnitude) of a vector is written with two vertical bars, denoted |a|. If a = (x, y), then its magnitude is |a| = sqrt(x^2 + y^2), which is exactly Pythagoras’ theorem applied in the coordinate plane: the horizontal and vertical components form the two legs of a right-angled triangle, and the vector itself is the hypotenuse. For example, a = (3, 4) gives |a| = sqrt(9 + 16) = 5.

模与数乘有一个重要关系：|ka| = |k| × |a|。也就是说，把向量缩放 k 倍，它的长度就缩放 |k| 倍。注意这里取的是 k 的绝对值：k = -2 时，方向反转但长度变为原来的 2 倍。例如 a = (3, 4) 的模是 5，那么 |-2a| = |-2| × 5 = 10，检验：(-2)a = (-6, -8)，模 = sqrt(36 + 64) = 10，结果一致。

Magnitude and scalar multiplication satisfy the important relation |ka| = |k| x |a|. In words, scaling a vector by k scales its length by |k|. Note the absolute value: when k = -2 the direction reverses but the length becomes twice the original. For example, a = (3, 4) has magnitude 5, so |-2a| = |-2| x 5 = 10; checking: (-2)a = (-6, -8) has magnitude sqrt(36 + 64) = 10, which matches.

单位向量（unit vector）是模为 1 的向量。任何非零向量 a 都可以通过数乘变成单位向量：单位向量 = (1 / |a|) × a。例如 a = (3, 4)，|a| = 5，单位向量为 (3/5, 4/5) = (0.6, 0.8)，它的模等于 1。单位向量的作用是指明方向：去掉长度信息，只保留方向。IGCSE 中单位向量偶尔出现在难题的铺垫部分，理解”除以模”的操作即可。

A unit vector is a vector with magnitude 1. Every non-zero vector a can be turned into a unit vector by scalar multiplication: unit vector = (1 / |a|) x a. For example, a = (3, 4) has |a| = 5, so the unit vector is (3/5, 4/5) = (0.6, 0.8), whose magnitude is 1. The role of a unit vector is to indicate direction: it strips away the length information and keeps only the direction. Unit vectors occasionally appear in the scaffolding of harder IGCSE questions; understanding the “divide by the magnitude” operation is sufficient.

八、位置向量与数乘：从原点出发的向量 | Position Vectors and Scalar Multiplication

位置向量（position vector）是指从原点 O 指向某一点的向量。点 P 的位置向量通常记作 p 或 OP。例如点 P(2, 5) 的位置向量就是 p = (2, 5)。位置向量把”点”和”向量”统一起来：一个点对应唯一的位置向量，反之亦然。这是向量方法能够解决几何问题的关键桥梁。

A position vector is the vector from the origin O to a given point. The position vector of point P is usually written p or OP. For example, the position vector of point P(2, 5) is p = (2, 5). Position vectors unify “points” and “vectors”: each point corresponds to exactly one position vector and vice versa. This is the key bridge that allows vector methods to solve geometric problems.

有了位置向量，任意两点间的向量可以简洁地表示：AB = b – a。这个公式非常常用。如果题目给出 A(1, 2) 和 B(4, 6)，则 AB = (4 – 1, 6 – 2) = (3, 4)。进一步，如果 M 是 AB 的中点，那么 M 的位置向量 m = (a + b) / 2 = (1/2)a + (1/2)b。这里就出现了数乘：中点位置向量是两个端点位置向量各取一半后相加。

With position vectors, the vector between any two points can be written concisely: AB = b – a. This formula is used constantly. If A(1, 2) and B(4, 6) are given, then AB = (4 – 1, 6 – 2) = (3, 4). Furthermore, if M is the midpoint of AB, the position vector of M is m = (a + b) / 2 = (1/2)a + (1/2)b. Scalar multiplication appears here: the midpoint position vector is half of each endpoint’s position vector, added together.

用分量验证中点公式：m = (1/2)(x1 + x2, y1 + y2)，这正是我们在坐标几何中学过的中点公式 ((x1 + x2)/2, (y1 + y2)/2)。向量方法和坐标方法在这里殊途同归。记住这个联系，考试中遇到”用向量证明 M 是 AB 的中点”时，只需要证明 m = (1/2)(a + b)，或者证明 AM = MB 且 A、M、B 共线。

Verifying the midpoint formula with components: m = (1/2)(x1 + x2, y1 + y2), which is exactly the midpoint formula ((x1 + x2)/2, (y1 + y2)/2) learned in coordinate geometry. The vector method and the coordinate method reach the same destination by different routes. Remember this link: when a question asks you to prove that M is the midpoint of AB using vectors, it suffices to show m = (1/2)(a + b), or to show that AM = MB and that A, M, B are collinear.

九、数乘在几何证明中的应用：中点、分点与共线 | Applications in Geometry Proofs: Midpoints, Dividing Points and Collinearity

Edexcel IGCSE 向量大题的经典套路是：给出一个三角形或四边形，标出若干中点或比例分点，要求证明某两条线段平行或某三点共线，最后求某个向量的表达式。这类题的核心工具就是数乘。例如：三角形 OAB 中，C 是 OA 的中点，D 是 OB 上满足 OD = 2DB 的点，则 OC = (1/2)a，OD = (2/3)b，于是 CD = OD – OC = (2/3)b – (1/2)a。

The classic pattern of Edexcel IGCSE vector questions is: a triangle or quadrilateral is given with several midpoints or proportional dividing points marked; you are asked to prove that two segments are parallel, or that three points are collinear, and finally to express a certain vector. The core tool in these questions is scalar multiplication. For example, in triangle OAB, C is the midpoint of OA and D is the point on OB with OD = 2DB; then OC = (1/2)a and OD = (2/3)b, so CD = OD – OC = (2/3)b – (1/2)a.

分点的比例要格外小心。OD = 2DB 意味着 D 把 OB 分成 2:1，所以 OD 占全长的 2/3，而不是 2/1 或 1/2。一个可靠的检查方法：如果 D 更靠近 B，那么 OD 应该接近全长，即系数接近 1。OD = (2/3)b 说明 D 在 OB 的 2/3 处，确实更靠近 B，与条件 OD = 2DB 一致。

Be very careful with the ratio of dividing points. OD = 2DB means D divides OB in the ratio 2:1, so OD is 2/3 of the whole length, not 2/1 or 1/2. A reliable check: if D is closer to B, then OD should be close to the whole length, so the coefficient should be close to 1. OD = (2/3)b places D at two-thirds of the way along OB, indeed closer to B, which agrees with the condition OD = 2DB.

证明共线的标准格式：先分别写出两个向量的表达式（通常共用一个起点），例如从 O 出发的 OX 和 OY；然后说明 OY = k × OX（k 为某个常数）；最后下结论：因为 OY 是 OX 的数乘，两向量平行，且它们都经过点 O，所以 O、X、Y 三点共线。注意：仅仅平行还不够，必须说明它们共起点（或共用一个公共点），才能推出三点共线。

The standard format for proving collinearity: first write the expressions of the two vectors (usually sharing a common starting point), for example OX and OY from O; then show that OY = k x OX for some constant k; finally conclude: since OY is a scalar multiple of OX, the two vectors are parallel, and since they both pass through O, the points O, X and Y are collinear. Note that parallelism alone is not enough – you must also point out that they share a common point (or a common start) before concluding the three points are collinear.

十、向量的模与数乘的结合：|ka| 的计算 | Combining Magnitude and Scalar Multiplication: Computing |ka|

有些题目直接给出向量的分量，要求计算缩放后的模。两步走：第一步，用数乘算出新向量的分量；第二步，用勾股定理算模。例如，a = (-3, 4)，求 |3a|。先算 3a = (-9, 12)，再算模 = sqrt(81 + 144) = sqrt(225) = 15。也可以直接用公式 |ka| = |k| × |a| = 3 × 5 = 15，两种方法结果一致，第二种更快。

Some questions give the components of a vector and ask you to compute the magnitude after scaling. Two steps: first, use scalar multiplication to find the components of the new vector; second, apply Pythagoras’ theorem to find the magnitude. For example, a = (-3, 4), find |3a|. First compute 3a = (-9, 12), then the magnitude = sqrt(81 + 144) = sqrt(225) = 15. Alternatively use the formula |ka| = |k| x |a| = 3 x 5 = 15; both methods agree, and the second is faster.

如果题目要求”求与 a 同方向、长度为某个值的向量”，那么思路是：先求单位方向 (1/|a|)a，再乘以目标长度。例如，求与 a = (6, 8) 同方向且长度为 2 的向量：|a| = 10，单位向量 = (0.6, 0.8)，目标向量 = 2 × (0.6, 0.8) = (1.2, 1.6)。这类问题把数乘、模和单位向量三个知识点串在一起，是综合题的热门素材。

If the question asks for “a vector in the same direction as a with a given length”, the idea is: first find the unit direction (1/|a|)a, then multiply by the target length. For example, find the vector in the same direction as a = (6, 8) with length 2: |a| = 10, the unit vector = (0.6, 0.8), and the target vector = 2 x (0.6, 0.8) = (1.2, 1.6). This type of question connects scalar multiplication, magnitude and unit vectors in one chain, making it popular material for combined questions.

在物理背景的应用题中也会出现数乘：力 F 的方向不变、大小变为 3 倍，就是 3F；速度反向且大小减半，就是 (-1/2)v。把物理语言翻译成向量语言时，注意”反向”对应负标量，”大小变为 n 倍”对应乘以 n。这种翻译能力在跨学科题目中是得分关键。

Scalar multiplication also appears in physics-context application questions: a force F keeping its direction with triple magnitude is 3F; a velocity reversed and halved is (-1/2)v. When translating physical language into vector language, note that “reversed” corresponds to a negative scalar and “magnitude becomes n times” corresponds to multiplying by n. This translation skill is the key to scoring in cross-discipline questions.

十一、常见考试题型与易错点清单 | Typical Exam Question Types and a Checklist of Common Mistakes

Edexcel IGCSE 关于数乘的常见题型可以归纳为五类。第一类：给出向量分量，直接计算 ka 或化简组合表达式。第二类：判断两个向量是否平行（用比值检验）。第三类：在几何图形中，用位置向量表示中点、分点间的向量。第四类：证明三点共线或两条线段平行。第五类：求缩放后向量的模或构造指定长度的同向向量。

The common Edexcel IGCSE question types on scalar multiplication can be summarised in five categories. Type 1: given the components, compute ka directly or simplify a combined expression. Type 2: decide whether two vectors are parallel (using the ratio test). Type 3: in a geometric figure, express the vector between midpoints or dividing points in terms of position vectors. Type 4: prove three points are collinear or two segments are parallel. Type 5: find the magnitude of a scaled vector, or construct a same-direction vector of a given length.

高频易错点第一号：数乘时只乘了一个分量。例如把 2(3, -4) 写成 (6, -4)。检查习惯：数乘后括号内必须仍然是两个数，且都与原向量成同一比例。第二号：分点比例用错，如把 OD = 2DB 写成 OD = (1/2)b。第三号：列向量上下颠倒。第四号：负标量方向判断错误，k 小于 0 时方向反转 180 度。第五号：模的计算中漏掉绝对值，|(-2)a| 的结果一定是正数。

Common mistake number one: multiplying only one component during scalar multiplication, for example writing 2(3, -4) as (6, -4). A checking habit: after scalar multiplication the bracket must still contain two numbers, both scaled by the same ratio as the original vector. Mistake two: using the wrong dividing ratio, such as writing OD = (1/2)b for OD = 2DB. Mistake three: swapping the rows of a column vector. Mistake four: judging the direction of a negative scalar wrongly – when k is less than 0 the direction reverses through 180 degrees. Mistake five: dropping the absolute value when computing a magnitude – |(-2)a| must always be positive.

最后一条考试策略：向量题永远要写出完整的表达式再代入数字。很多同学喜欢心算，但 Edexcel 的评分标准（mark scheme）通常会给”方法分”（method marks）：即使最后答案算错，只要表达式、平行关系或共线结论的推导过程正确，仍然能拿到大部分分数。所以过程要写清楚，特别是”因为 OY = 2OX，所以 O、X、Y 共线”这样的关键句不能省略。

One final exam strategy: in vector questions always write down the complete expression before substituting numbers. Many students prefer mental arithmetic, but the Edexcel mark scheme usually awards method marks: even if the final answer is wrong, you still earn most of the marks as long as the working – the expression, the parallelism relation, or the collinearity deduction – is correct. So write out the process clearly, and never omit key sentences such as “since OY = 2OX, the points O, X and Y are collinear”.

十二、练习与详细解析 | Practice Questions with Worked Solutions

练习一：已知 a = (2, -5)，求 3a 和 -2a。解析：3a = (6, -15)，-2a = (-4, 10)。两个分量都要乘以标量，负标量会把两个分量的符号都反过来。

Practice 1: Given a = (2, -5), find 3a and -2a. Solution: 3a = (6, -15) and -2a = (-4, 10). Both components must be multiplied by the scalar, and a negative scalar flips the sign of both components.

练习二：判断向量 p = (4, -6) 与 q = (-2, 3) 是否平行。解析：比值 k1 = -2/4 = -0.5，k2 = 3/(-6) = -0.5，两个比值相等，所以 q = (-0.5)p，两向量平行且方向相反。注意两个比值都是负的，说明 k 是负数，方向相反。

Practice 2: Decide whether vectors p = (4, -6) and q = (-2, 3) are parallel. Solution: ratio k1 = -2/4 = -0.5 and ratio k2 = 3/(-6) = -0.5; the two ratios are equal, so q = (-0.5)p, meaning the vectors are parallel and point in opposite directions. Note that both ratios are negative, so k is negative and the directions are opposite.

练习三：点 A(1, 2)、B(5, 10)，M 是 AB 的中点，求 M 的坐标。解析：m = (1/2)(a + b) = (1/2)((1, 2) + (5, 10)) = (1/2)(6, 12) = (3, 6)。数乘 (1/2) 把两个分量同时减半。检验：从 A 到 M 是 (2, 4)，从 M 到 B 也是 (2, 4)，确实等距且共线。

Practice 3: Points A(1, 2) and B(5, 10) are given, and M is the midpoint of AB. Find the coordinates of M. Solution: m = (1/2)(a + b) = (1/2)((1, 2) + (5, 10)) = (1/2)(6, 12) = (3, 6). The scalar (1/2) halves both components at the same time. Check: from A to M is (2, 4) and from M to B is also (2, 4), so the distances are equal and the points are collinear.

练习四：已知 a = (-4, 3)，求与 a 同方向且长度为 5 的向量。解析：|a| = sqrt(16 + 9) = 5，巧合的是模正好等于 5，所以目标向量就是 a 本身 = (-4, 3)。如果目标长度改为 10，则目标向量 = (10/5) × (-4, 3) = (-8, 6)。关键步骤是先用模求出比例系数 k = 目标长度 / |a|。

Practice 4: Given a = (-4, 3), find the vector in the same direction as a with length 5. Solution: |a| = sqrt(16 + 9) = 5; coincidentally the magnitude is exactly 5, so the target vector is a itself = (-4, 3). If the target length were 10, the target vector would be (10/5) x (-4, 3) = (-8, 6). The key step is to find the scaling factor k = target length / |a| using the magnitude first.

练习五：三角形 OAB 中，a = OA，b = OB，点 C 在 AB 上且 AC = CB，点 D 在 OB 上且 OD = (2/3)OB。用 a 和 b 表示 CD，并判断 CD 是否平行于 OA。解析：AC = CB 说明 C 是 AB 的中点，所以 OC = (1/2)(a + b)；OD = (2/3)b；于是 CD = OD – OC = (2/3)b – (1/2)(a + b) = (2/3)b – (1/2)a – (1/2)b = (1/6)b – (1/2)a。CD 中同时含有 a 和 b 的项，不是 a 的纯倍数，所以 CD 不平行于 OA。

Practice 5: In triangle OAB, a = OA and b = OB. Point C lies on AB with AC = CB, and point D lies on OB with OD = (2/3)OB. Express CD in terms of a and b, and decide whether CD is parallel to OA. Solution: AC = CB means C is the midpoint of AB, so OC = (1/2)(a + b); OD = (2/3)b; therefore CD = OD – OC = (2/3)b – (1/2)(a + b) = (2/3)b – (1/2)a – (1/2)b = (1/6)b – (1/2)a. Since CD contains terms in both a and b, it is not a pure multiple of a, so CD is not parallel to OA.

Summary | 总结

本文系统梳理了 IGCSE Edexcel 数学中向量的数乘运算：从向量的定义与列向量表示出发，介绍了数乘的运算法则 ka = (kx, ky) 及其几何意义（伸缩、反向、零向量），并重点讲解了数乘在平行判定、单位向量、位置向量、中点公式和共线证明中的应用。每一条规则都配了具体例题，最后给出了五道带解析的练习题和易错点清单。

This article systematically reviews scalar multiplication of vectors in IGCSE Edexcel Mathematics: starting from the definition of vectors and column vector notation, it introduces the rule ka = (kx, ky) and its geometric meaning (stretching, reversing and the zero vector), with particular attention to its applications in parallelism tests, unit vectors, position vectors, the midpoint formula and collinearity proofs. Every rule is accompanied by concrete examples, and the article closes with five practice questions with worked solutions and a checklist of common mistakes.

数乘的本质是”按比例缩放并可选地反转方向”。掌握了数乘，就掌握了向量章节的钥匙：平行、共线、中点、分点这些高频考点全部建立在”一个向量是另一个向量的数倍”这个核心思想上。建议同学们在复习时把本文的练习题独立重做一遍，并用”先写表达式、再代入、最后用比值检验”的三步法检查每一道向量题。

The essence of scalar multiplication is “scaling by a ratio, with an optional reversal of direction”. Master scalar multiplication and you hold the key to the whole vectors chapter: parallelism, collinearity, midpoints and dividing points – all the high-frequency examination topics – rest on the core idea that “one vector is a scalar multiple of another”. When revising, we recommend redoing the practice questions in this article independently, and checking every vector question with the three-step method: write the expression first, then substitute, and finally verify with the ratio test.

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