📚 Vectors | 向量
Vectors are a fundamental part of IGCSE Edexcel Mathematics. They describe quantities that have both magnitude and direction, unlike scalars which only have magnitude. Understanding vectors is essential for solving problems in coordinate geometry, mechanics, and many real-life applications. This article covers all key concepts, from notation to geometric proofs, with clear English and Chinese explanations side by side.
向量是 IGCSE Edexcel 数学中的一个基础主题。它们描述既有大小又有方向的量，与仅有大小的标量不同。理解向量对于解决坐标几何、力学以及许多实际应用问题至关重要。本文涵盖从记法到几何证明的所有关键概念，并配有清晰的中英双语解释。
1. Introduction to Vectors | 向量简介
A vector is a quantity defined by its magnitude (size) and direction. Examples include displacement, velocity, and force. A scalar has only magnitude, for instance, distance, speed, and mass. In written form, vectors are often denoted by bold letters (e.g. a) or with an arrow above the letter (e.g. a⃗).
向量是由大小（模）和方向定义的量。例如位移、速度和力。标量只有大小，比如距离、速率和质量。在书面形式中，向量通常用粗体字母（如 a）或在字母上方加箭头（如 a⃗）来表示。
2. Vector Notation and Representation | 向量的记法与表示
Vectors can be represented as a column vector, such as a x y ) (written in one column: top component x, bottom component y), or using unit vectors i and j in two dimensions. For example, a = xi + yj. The magnitude of a vector a is denoted |a|.
向量可以表示为列向量，如 a = ( x y ) （写成一列：上分量 x，下分量 y），或使用二维中的单位向量 i 和 j。例如 a = xi + yj。向量 a 的模记作 |a|。
3. Equality of Vectors | 向量相等
Two vectors are equal if they have the same magnitude and direction, regardless of their initial point. For column vectors, a = ( x₁ y₁ ) equals b = ( x₂ y₂ ) if and only if x₁ = x₂ and y₁ = y₂.
两个向量如果具有相同的大小和方向，则相等，无论它们的起点在哪里。对于列向量，a = ( x₁ y₁ ) 等于 b = ( x₂ y₂ ) 当且仅当 x₁ = x₂ 且 y₁ = y₂。
4. Addition of Vectors | 向量加法
Vectors are added component-wise. If a = ( x₁ y₁ ) and b = ( x₂ y₂ ), then a + b = ( x₁ + x₂ y₁ + y₂ ). Geometrically, place the tail of b at the head of a; the resultant vector goes from the tail of a to the head of b. This is the triangle law of addition.
向量按对应分量相加。若 a = ( x₁ y₁ ) 且 b = ( x₂ y₂ )，则 a + b = ( x₁ + x₂ y₁ + y₂ )。几何上，将 b 的尾部放在 a 的头部；合向量从 a 的尾部指向 b 的头部。这就是加法的三角形法则。
5. Subtraction of Vectors | 向量减法
Subtracting b from a is equivalent to adding the negative of b: a − b = a + (−b). The negative of a vector has the same magnitude but opposite direction. Component-wise: ( x₁ − x₂ y₁ − y₂ ).
从 a 减去 b 相当于加上 b 的负向量：a − b = a + (−b)。负向量具有相同的大小，但方向相反。按分量运算为：( x₁ − x₂ y₁ − y₂ )。
6. Multiplication by a Scalar | 标量乘法
If a is a vector and k is a scalar, then ka is a vector parallel to a. Its magnitude is |k| times the magnitude of a, and its direction is the same as a if k > 0, opposite if k < 0. For a column vector a = ( x y ), ka = ( kx ky ).
若 a 是向量，k 是标量，则 ka 是一个与 a 平行的向量。其大小为 |a| 的 |k| 倍，方向在 k > 0 时与 a 相同，在 k < 0 时相反。对于列向量 a = ( x y )，ka = ( kx ky )。
7. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector a = ( x y ) is given by
|a| = √x² + y²
This comes from the Pythagorean theorem. For a = xi + yj, the magnitude is the same expression. The magnitude is always a non-negative scalar.
向量 a = ( x y ) 的模（长度）由下式给出：
|a| = √x² + y²
这来自勾股定理。对于 a = xi + yj，模的表达式相同。模总是非负标量。
8. Position Vectors | 位置向量
A position vector represents the location of a point relative to an origin O. If point P has coordinates (x, y), its position vector is OP = ( x y ) = xi + yj. The vector between two points A and B with position vectors a and b is AB = b − a.
位置向量表示一个点相对于原点 O 的位置。若点 P 的坐标为 (x, y)，则其位置向量为 OP = ( x y ) = xi + yj。具有位置向量 a 和 b 的两点 A 和 B 之间的向量为 AB = b − a。
9. Vector Geometry | 向量几何
Vectors can be used to express the sides and diagonals of geometric shapes. For example, in a parallelogram with vertices O, A, B, and C such that OA = a and OB = b, the diagonal OC = a + b, and the other diagonal AB = b − a. Midpoint of AB has position vector ½(a + b).
向量可用于表达几何图形的边和对角线。例如，在平行四边形中，设顶点 O, A, B, C 满足 OA = a 且 OB = b，则对角线 OC = a + b，另一条对角线 AB = b − a。AB 的中点具有位置向量 ½(a + b)。
10. Collinearity and Parallel Vectors | 共线与平行向量
Two vectors are parallel if one is a scalar multiple of the other: b = ka for some scalar k. Three points A, B, C are collinear (lie on the same straight line) if the vectors AB and AC are parallel, which means AB = k AC for some scalar k. This implies the points share a common direction vector and are connected.
如果两个向量互为标量倍数，即存在某个标量 k 使得 b = ka，则它们平行。三点 A, B, C 共线（位于同一直线上）当且仅当向量 AB 与 AC 平行，即存在某个标量 k 使得 AB = k AC。这意味着这些点共享一个共同的方向向量并且相互连接。
11. Solving Vector Problems | 解决向量问题
Typical IGCSE vector questions involve finding an unknown vector in terms of given vectors, proving that lines are parallel or that points are collinear, or finding ratios on line segments. Start by clearly defining all known vectors in terms of positions or given directions. Express the target vector as a combination of known vectors using path rules (e.g., AB = AO + OB = −OA + OB). Then simplify and compare with given conditions.
典型的 IGCSE 向量问题包括用给定向量表示未知向量、证明两直线平行或证明三点共线，或求线段上的比例。首先用位置向量或已知方向明确定义所有已知向量。使用路径规则（如 AB = AO + OB = −OA + OB）将目标向量表示为已知向量的组合。然后化简并与给定条件进行比较。
For example, to find the vector XY given OX = 2a + 3b and OY = 5a − b: XY = OY − OX = (5a − b) − (2a + 3b) = 3a − 4b.
例如，已知 OX = 2a + 3b 和 OY = 5a − b，求向量 XY：XY = OY − OX = (5a − b) − (2a + 3b) = 3a − 4b。
12. Summary | 总结
Vectors provide a powerful language for describing positions, movements, and geometric relationships. Mastering vector addition, subtraction, scalar multiplication, and the concept of parallel and collinear vectors is key to scoring well in this topic. Always think in terms of both magnitude and direction, and practice translating geometric diagrams into vector equations.
向量为描述位置、运动和几何关系提供了一种强大的语言。掌握向量的加法、减法、标量乘法以及平行与共线向量的概念是学好这一主题的关键。始终从大小和方向两个角度思考，并练习将几何图形转化为向量方程。
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