📚 Vectors: Representation, Addition and Scalar Multiplication | 向量：表示、加减与数乘
In IGCSE Mathematics, vectors open the door to describing both magnitude and direction simultaneously. They are essential for solving problems involving movement, forces, and geometry. This article covers the fundamentals of vector representation, column vectors, addition, subtraction, and multiplication by a scalar, all tailored to the Edexcel IGCSE syllabus.
在 IGCSE 数学中，向量打开了同时描述大小和方向的大门。它们对于解决涉及运动、力和几何的问题至关重要。本文涵盖向量的基本表示、列向量、加法、减法和数乘，全部针对 Edexcel IGCSE 教学大纲。
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, displacement of 5 km north is a vector, while mass of 10 kg is a scalar.
向量是既有大小又有方向的量。相比之下，标量只有大小。例如，向北位移 5 公里是向量，而 10 千克的质量是标量。
Vectors are typically represented by a directed line segment with an arrow. The length of the line shows the magnitude, and the arrowhead shows the direction. In print, vectors are often bolded (e.g., a) or written with a bar underneath, while in handwriting we place an arrow above the letter or underline it.
向量通常用带箭头的有向线段表示。线段的长度表示大小，箭头表示方向。在印刷体中，向量常被加粗（如 a），或者在下加横线；手写时，我们在字母上方画箭头或下划线。
The magnitude of vector a is written as |a|. Two vectors are equal if they have the same magnitude and the same direction, regardless of their starting points.
向量 a 的大小记作 |a|。如果两个向量大小相等且方向相同，则它们相等，无论起点位置如何。
2. Column Vectors | 列向量
A convenient way to describe a vector in 2D is by using a column vector. This gives the horizontal (x) and vertical (y) components. For example, a vector that goes 3 units to the right and 4 units up is written as:
描述二维向量的一种便捷方法是使用列向量。它给出水平 (x) 和垂直 (y) 分量。例如，向右 3 个单位、向上 4 个单位的向量写作：
(3, 4)ᵀ or 3 above 4 as a column, but here we use: [ 3 over 4 ] in text, but HTML render:
We will represent it as a column:
v =
3
4 (in a 2×1 matrix)
Actually, to display properly without LaTeX, we can use:
v = (3, 4)ᵀ means transposed, or we can write as a matrix using a table or simple text alignment. Let's do:
v = 3
4 in one line? Better to use Unicode brackets: ⎡3⎤ not standard. So I'll describe column vector explicitly and use bold: v = 3, 4 with explanation that the first number is the x-component, second is y. But we can use a simple table. For clarity, I'll use a table with no borders to mimic column vector.
Instead, I'll adopt the common notation: v = 3
4 as a column. More precisely:
A column vector is written as (x, y) ge parenthesis, but since we cannot use LaTeX, I will write it as a horizontal pair with a 'column vector' descriptor, and show the vertical arrangement using HTML table with border="0" and small padding:
Vector v =
| 3 |
| 4 |
A column vector is written as
| x |
| y |
列向量写作
| x |
| y |
Example: The vector from point A to point B with coordinates (2, 3) to (5, 7) is
| 5-2 |
| 7-3 |
| 3 |
| 4 |
示例：从点 A (2, 3) 到点 B (5, 7) 的向量为
| 5-2 |
| 7-3 |
| 3 |
| 4 |
3. Magnitude of a Vector | 向量的大小
For a column vector
| x |
| y |
对于列向量
| x |
| y |
Example: The magnitude of
| 3 |
| 4 |
示例：
| 3 |
| 4 |
Magnitude is always non-negative and represents the length of the directed line segment.
大小总是非负的，表示有向线段的长度。
4. Zero and Unit Vectors | 零向量与单位向量
The zero vector, denoted by 0 =
| 0 |
| 0 |
零向量，记作 0 =
| 0 |
| 0 |
A unit vector has magnitude exactly 1. For any non-zero vector v, the unit vector in the same direction is v / |v|.
单位向量的大小恰好为 1。对于任何非零向量 v，与其同方向的单位向量为 v / |v|。
The standard basis vectors in 2D are i =
| 1 |
| 0 |
| 0 |
| 1 |
| x |
| y |
二维空间的标准基底向量为 i =
| 1 |
| 0 |
| 0 |
| 1 |
| x |
| y |
5. Vector Addition | 向量加法
Vectors are added component-wise. If a =
| x₁ |
| y₁ |
| x₂ |
| y₂ |
| x₁ + x₂ |
| y₁ + y₂ |
向量按分量相加。如果 a =
| x₁ |
| y₁ |
| x₂ |
| y₂ |
| x₁ + x₂ |
| y₁ + y₂ |
Geometrically, vector addition follows the triangle law: place the tail of b at the head of a, then the resultant vector goes from the tail of a to the head of b. Alternatively, the parallelogram law can be used when the vectors share the same initial point.
几何上，向量加法遵循三角形法则：将 b 的尾部置于 a 的头部，则合向量从 a 的尾部指向 b 的头部。或者，当向量起点相同时，可使用平行四边形法则。
Addition is commutative: a + b = b + a, and associative: (a + b) + c = a + (b + c).
加法满足交换律：a + b = b + a，以及结合律：(a + b) + c = a + (b + c)。
Example: Given p =
| 2 |
| -1 |
| -3 |
| 5 |
| 2 + (-3) |
| -1 + 5 |
| -1 |
| 4 |
示例：给定 p =
| 2 |
| -1 |
| -3 |
| 5 |
| 2 + (-3) |
| -1 + 5 |
| -1 |
| 4 |
6. Vector Subtraction | 向量减法
To subtract b from a, we add the negative of b: a - b = a + (-b). The negative of a vector has the same magnitude but opposite direction.
要从 a 中减去 b，我们加上 b 的负向量：a - b = a + (-b)。向量的负向量大小相同但方向相反。
In column form, if b =
| x |
| y |
| -x |
| -y |
| x₁ - x₂ |
| y₁ - y₂ |
以列向量形式，如果 b =
| x |
| y |
| -x |
| -y |
| x₁ - x₂ |
| y₁ - y₂ |
Geometrically, the vector a - b can be represented by the diagonal from the head of b to the head of a when they share a common tail.
几何上，向量 a - b 可用当 a 和 b 共起点时，从 b 头部指向 a 头部的对角线表示。
Example:
| 5 |
| 2 |
| 1 |
| -3 |
| 4 |
| 5 |
示例：
| 5 |
| 2 |
| 1 |
| -3 |
| 4 |
| 5 |
7. Scalar Multiplication | 标量乘法
When a vector is multiplied by a scalar (a real number), each component is multiplied by that scalar. If v =
| x |
| y |
| kx |
| ky |
当向量乘以一个标量（一个实数）时，每个分量都乘以这个标量。如果 v =
| x |
| y |
| kx |
| ky |
The magnitude scales by the absolute value: |kv| = |k| |v| remains the same if k > 0, and reverses if k < 0. If k = 0, the result is the zero vector.
大小按绝对值缩放：|kv| = |k| |v|。如果 k > 0，方向不变；如果 k < 0，方向相反。如果 k = 0，结果为零向量。
Example: If a =
| 2 |
| -1 |
| 6 |
| -3 |
| -4 |
| 2 |
示例：如果 a =
| 2 |
| -1 |
| 6 |
| -3 |
| -4 |
| 2 |
Scalar multiplication distributes over addition: k(a + b) = ka + kb.
标量乘法对加法满足分配律：k(a + b) = ka + kb。
8. Position Vectors | 位置向量
A position vector of a point P is the vector from the origin O to P. If P has coordinates (x, y), its position vector is OP =
| x |
| y |
点 P 的位置向量是从原点 O 到 P 的向量。如果 P 的坐标为 (x, y)，则其位置向量为 OP =
| x |
| y |
Using position vectors, the vector from point A to point B can be expressed as OB - OA. This is a powerful tool for solving coordinate geometry problems with vectors.
利用位置向量，从点 A 到点 B 的向量可表示为 OB - OA。这是用向量解决坐标几何问题的强大工具。
Example: If A = (1, 2) and B = (4, 6), then AB =
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