Vectors | 向量

📚 Vectors | 向量

Vectors are quantities that describe both a magnitude and a direction. In the IGCSE Edexcel Mathematics syllabus, vectors are essential for solving problems in geometry, transformations, and coordinate systems. You must be confident in reading, writing, manipulating, and interpreting vectors.

向量是同时描述“大小”和“方向”的量。在 Edexcel IGCSE 数学考纲中，向量是解决几何、变换和坐标系问题的重要工具。你必须熟练掌握向量的读法、写法、运算以及几何意义。

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1. What is a Vector? | 什么是向量？

A scalar is a quantity that has only magnitude, such as speed, mass, or temperature. A vector has both magnitude and direction, such as displacement, velocity, or force. The length of a vector represents its magnitude, and the arrow indicates its direction.

标量（scalar）是只有大小的量，比如速率、质量或温度。向量（vector）同时具有大小和方向，比如位移、速度或力。向量的长度表示它的大小，箭头表示它的方向。

• A vector written as AB means the vector from point A to point B.

  AB 表示从点 A 指向点 B 的向量。

• Vectors can be written in bold, such as v, or with an arrow, such as v⃗.

  向量可以用粗体如 v，或带箭头如 v⃗ 来表示。

• In printed materials, the vector AB is often written as AB with a line underneath, but arrow notation is also common.

  在印刷资料中，向量 AB 常用下划线或箭头符号表示，两种方式都常见。

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2. Vector Notation | 向量的表示方法

In IGCSE, you are expected to recognise different forms of vector notation. The column vector is the most common algebraic form.

在 IGCSE 中，你需要辨认不同的向量记号。列向量是最常见的代数形式。

Name | 名称	Notation | 记号
Column vector	( 35 )
Unit vector notation	a = x i + y j
Letter with arrow	v⃗
Magnitude	|v| or |AB|

The top number in a column vector is the x-component (horizontal), and the bottom number is the y-component (vertical).

列向量中上方的数是 x 分量（水平方向），下方的数是 y 分量（垂直方向）。

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3. Column Vectors | 列向量

A column vector expresses the displacement from one point to another. For example, the column vector ( 4-2 ) means move 4 units right and 2 units down.

列向量表示从一个点到另一个点的位移。例如列向量 ( 4-2 ) 表示向右移动 4 个单位，向下移动 2 个单位。

• Positive x means right; negative x means left.

  x 为正表示向右，x 为负表示向左。

• Positive y means up; negative y means down.

  y 为正表示向上，y 为负表示向下。

If A has coordinates (1,3) and B has coordinates (5,1), then the vector AB = ( 5-11-3 ) = ( 4-2 ).

如果 A 的坐标为 (1,3)，B 的坐标为 (5,1)，则向量 AB = ( 5-11-3 ) = ( 4-2 )。

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4. Equal and Negative Vectors | 相等向量与负向量

Two vectors are equal if they have the same magnitude and the same direction. This means their column vector representations are identical, regardless of where they are drawn.

两个向量相等，当且仅当它们的大小和方向都相同。也就是说，它们的列向量表示完全一样，与所画的位置无关。

The negative of a vector has the same magnitude but the opposite direction. If v = ( ab ), then −v = ( -a-b ).

负向量与原来向量大小相同但方向相反。如果 v = ( ab )，那么 −v = ( -a-b )。

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5. Adding and Subtracting Vectors | 向量的加减法

To add two column vectors, add the x-components and add the y-components separately. For example:

两个列向量相加，只需分别相加 x 分量和 y 分量。例如：

( ab ) + ( cd ) = ( a+cb+d )

To subtract a vector, add its negative. So u − v = u + (−v). This is equivalent to subtracting components.

减去一个向量相当于加上它的负向量，即 u − v = u + (−v)。这等价于对应分量相减。

Geometrically, adding vectors follows the triangle law: if you place the tail of the second vector at the head of the first, the sum is the vector from the tail of the first to the head of the second.

几何上，向量加法遵循三角形法则：将第二个向量的起点放在第一个向量的终点，则和向量是从第一个向量起点指向第二个向量终点的向量。

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6. Multiplying a Vector by a Scalar | 向量的数乘

When a vector is multiplied by a scalar k, every component is multiplied by k. The magnitude of the vector is multiplied by |k|. If k is positive, the direction is unchanged; if k is negative, the direction is reversed.

当向量乘以标量 k 时，每个分量都乘以 k。向量的大小变为原来的 |k| 倍。如果 k 为正，方向不变；如果 k 为负，方向相反。

k × ( ab ) = ( kakb )

For example, 2 × ( 3-1 ) = ( 6-2 ).

例如，2 × ( 3-1 ) = ( 6-2 )。

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7. Magnitude of a Vector | 向量的模

The magnitude (length) of a column vector ( xy ) is found using Pythagoras’ theorem:

列向量 ( xy ) 的模（长度）使用勾股定理计算：

|v| = √(x² + y²)

For example, the magnitude of ( 34 ) is √(3² + 4²) = √(9 + 16) = √25 = 5.

例如，( 34 ) 的模为 √(3² + 4²) = √(9 + 16) = √25 = 5。

• The magnitude is always positive.

  模始终为正数。

• If the magnitude is 1, the vector is called a unit vector.

  如果模为 1，则称该向量为单位向量。

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8. Parallel Vectors | 平行向量

Two vectors are parallel if one is a scalar multiple of the other. That is, a and b are parallel if there exists a real number k such that a = kb.

两个向量平行，当且仅当其中一个向量是另一个向量的标量倍。也就是说，a 和 b 平行，如果存在实数 k，使得 a = kb。

Example: Are ( 24 ) and ( 612 ) parallel? Yes, because ( 612 ) = 3 × ( 24 ).

例：( 24 ) 与 ( 612 ) 是否平行？是，因为 ( 612 ) = 3 × ( 24 )。

In geometry, this test is useful to prove that two line segments are parallel.

在几何中，这个判定方法常用于证明两条线段平行。

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