WJEC IGCSE Maths: Vectors Revision Guide | IGCSE WJEC 数学：向量考点精讲

📚 WJEC IGCSE Maths: Vectors Revision Guide | IGCSE WJEC 数学：向量考点精讲

A vector is a quantity that has both magnitude (size) and direction. In WJEC IGCSE Mathematics, vectors appear regularly on both calculator and non-calculator papers. They are used to describe movement, position, and geometric relationships. Mastering vector addition, scalar multiplication, and geometric proof will give you a reliable source of marks. This revision guide covers every essential topic from notation to exam-style proof questions, with clear explanations in English and Chinese.

向量是一种既有大小（模）又有方向的量。在 WJEC IGCSE 数学考试中，向量在计算器和非计算器试卷中都会出现，常用于描述移动、位置以及几何关系。熟练掌握向量的加法、标量乘法和几何证明题型，能帮你稳稳拿到分数。本文为你系统梳理向量的表示、运算、模长、位置向量以及几何应用，中英双语讲解，助你高效备考。

1. What is a Vector? | 什么是向量？

A vector is a quantity with both size (magnitude) and direction. Examples include displacement, velocity, and force. A quantity with only size is called a scalar, such as distance or speed. In diagrams, vectors are drawn as arrows, where the length shows magnitude and the arrowhead shows direction. Vectors are usually named with a bold lowercase letter, such as a, or with an arrow above, like \vec{a}. In WJEC IGCSE, you will see both notations.

向量是既有大小又有方向的量，如位移、速度和力。只有大小的量称为标量，比如距离或速率。在图形中，向量用带箭头的线段表示，线段长度代表模的大小，箭头指向代表方向。向量通常用粗体小写字母表示，如 a，或上方带箭头，如 \vec{a}。在 WJEC IGCSE 中这两种记法都会出现。

2. Representing Vectors: Column Vectors and Unit Vectors | 向量表示：列向量与单位向量

Vectors can be written as column vectors. For example, a vector with horizontal component 3 and vertical component 5 can be written as

a = (3 \\ 5)
where the top number is the x-component and the bottom number is the y-component. Another common representation uses the unit vectors i and j. The vector (3 \\ 5) can be expressed as 3i + 5j. WJEC papers use both forms; you must be comfortable converting between them. A column vector shows movement: right is positive, left is negative, up is positive, down is negative.

向量可以写成列向量的形式。例如，水平分量为 3，竖直分量为 5 的向量可以记作

a = (3 \\ 5)
其中上面的数字是 x 分量，下面的数字是 y 分量。另一种常见表示使用单位向量 i 和 j。向量 (3 \\ 5) 可以写成 3i + 5j。WJEC 试卷中两种形式都会出现，你必须能熟练转换。列向量描述移动：向右为正，向左为负；向上为正，向下为负。

3. Vector Addition and Subtraction | 向量的加法和减法

To add two vectors, add their corresponding components. If a = (x₁ \\ y₁) and b = (x₂ \\ y₂), then

a + b = (x₁ + x₂ \\ y₁ + y₂).
Subtraction works similarly: a – b = (x₁ – x₂ \\ y₁ – y₂).
Geometrically, adding vectors can be shown by placing the tail of the second vector at the head of the first. The resulting vector from start to finish is the sum. In term of unit vectors, (2i + 3j) + (5i – j) = 7i + 2j.

两个向量相加，只需将对应分量相加。如果 a = (x₁ \\ y₁) 且 b = (x₂ \\ y₂)，那么

a + b = (x₁ + x₂ \\ y₁ + y₂)。
减法是类似的：a – b = (x₁ – x₂ \\ y₁ – y₂)。
从几何角度看，向量加法可以通过将第二个向量的起点放在第一个向量的终点上来表示，从起点到终点的向量就是和向量。用单位向量表示，(2i + 3j) + (5i – j) = 7i + 2j。

4. Scalar Multiplication | 标量乘法（数乘）

When you multiply a vector by a scalar (a number), you multiply each component by that number. If a = (x \\ y), then k a = (k x \\ k y).
Geometrically, scalar multiplication changes the length of the vector without changing its direction (if k is positive). If k is negative, the direction reverses. For example, 3a is three times as long as a and points in the same direction, while -2a is twice as long and points in the opposite direction. This property is essential for proving parallel and collinear vectors.

当一个向量乘以标量（一个数）时，你需要把向量的每个分量都乘以这个数。如果 a = (x \\ y)，那么 k a = (k x \\ k y)。
从几何上看，标量乘法会改变向量的长度，但不会改变其方向（假设 k 为正）。如果 k 为负，方向则相反。例如，3a 的长度是 a 的三倍且方向相同；-2a 的长度是两倍且方向相反。这一性质是证明平行向量和共线向量的基础。

5. Magnitude (Length) of a Vector | 向量的大小（模）

The magnitude or length of a vector a = (x \\ y) is found using Pythagoras’ theorem:

|a| = √(x² + y²).
For a vector expressed as xi + yj, the magnitude is also √(x² + y²).
Magnitude is always a non-negative number. In WJEC questions, you may need to find the length of a side in a shape or the distance between two points by treating the path as a vector and calculating its magnitude. Always leave your answer in surd form unless asked for a decimal.

向量 a = (x \\ y) 的模长可以通过勾股定理求得：

|a| = √(x² + y²)。
对于表示为 xi + yj 的向量，模长同样是 √(x² + y²)。
模长永远是非负数。在 WJEC 考题中，你可能需要求图形中某条边的长度，或两点间的距离：把路径视为向量，再计算其模长。除非题目明确要求小数，否则保留根号形式。

6. Position Vectors and Displacement Vectors | 位置向量与位移向量

If you fix an origin O, the position vector of a point A is the vector OA, which gives the location of A relative to O. A displacement vector such as AB describes the movement from A to B. A key formula is

AB = OB – OA = b – a,
where a and b are the position vectors of A and B. This relation allows you to express any side of a triangle or quadrilateral in terms of given position vectors. WJEC problems often ask you to find an unknown vector in a diagram by building a route from one known vector to another.

一旦固定原点 O，点 A 的位置向量就是 OA，它表示 A 相对于 O 的位置。位移向量如 AB 则描述从 A 移动到 B 的过程。一个关键公式是

AB = OB – OA = b – a，
其中 a 和 b 分别为 A、B 的位置向量。利用这个关系，你可以将三角形或四边形的任一边用已知的位置向量表示出来。WJEC 的题目经常要求你通过构造从已知向量到未知向量的路径，求出图形中的某个向量。

7. Geometric Relationships: Parallel and Collinear Vectors | 几何关系：平行与共线向量

Two vectors a and b are parallel if one is a scalar multiple of the other: a = k b for some scalar k. This is a fundamental tool in vector proof.
Three points A, B, and C are collinear (lie on the same straight line) if the vectors AB and BC are parallel. In practice, you find AB and BC, then show that AB = k BC for some value of k. If the vectors share a common point, they are collinear. WJEC mark schemes expect you to state clearly that ‘vectors are parallel and share a point, therefore the points are collinear’.

如果其中一个向量是另一个向量的标量倍数，即 a = k b（k 为某标量），则这两个向量平行。这是向量证明的核心工具。
如果向量 AB 与 BC 平行，则三点 A、B、C 共线（位于同一直线上）。具体操作时，你求出 AB 和 BC，然后证明 AB = k BC 对某个 k 成立。如果两个向量有公共点，它们就是共线的。WJEC 评分标准要求你明确指出“向量平行且具有公共点，因此三点共线”。

8. Midpoints and Vector Proof | 中点与向量证明

If M is the midpoint of AB, then the position vector of M is

m = (a + b) / 2,
where a and b are the position vectors of A and B. This formula works with both column vectors and unit vector notation. In WJEC, you may be asked to prove that a quadrilateral is a parallelogram. For a parallelogram ABCD, diagonals bisect each other, so the midpoint of AC equals the midpoint of BD. Setting these equal yields a vector equation that you can solve. Always show your working step by step, expressing each vector in terms of a, b, etc., and use the scalar multiple condition to establish parallel sides.

如果 M 是 AB 的中点，则 M 的位置向量为

m = (a + b) / 2，
其中 a 和 b 是 A 和 B 的位置向量。这个公式既适用于列向量，也适用于单位向量表示。在 WJEC 考试中，你可能会被要求证明某四边形是平行四边形。对于平行四边形 ABCD，对角线互相平分，所以 AC 的中点等于 BD 的中点。令两者相等可得到一个向量方程并求解。解题时务必逐步展示过程，将每个向量用 a、b 等表示，并运用标量倍数条件证明对边平行。

9. Solving Geometric Problems with Vectors | 运用向量解决几何问题

Typical WJEC vector questions provide a diagram with some vectors labelled, e.g. AB = a, BC = b. You are asked to express other segments in terms of a and b. The trick is to trace a route from one point to another using known vectors. For instance, AD = AB + BD. If BD is not given directly, you may need to break it into steps involving midpoints, ratios, or parallel sides. Always pay attention to direction: BA = -a when AB = a. Ratio problems are common: if a point P divides a segment in the ratio m:n, its position vector can be found using section formula p = (n a + m b) / (m + n). Practice interpreting worded conditions into vector equations.

WJEC 的典型向量题会给出一个图形，并标注部分向量，如 AB = a，BC = b，然后要求你用 a 和 b 表示其他线段。技巧是利用已知向量规划出一条从一点到另一点的路径。例如 AD = AB + BD。如果 BD 未直接给出，你可能需要借助中点、比例或对边平行等条件进一步分解。始终注意方向：如果 AB = a，则 BA = -a。比例问题很常见：若点 P 以 m:n 的比例分割线段，则其位置向量可用定比分点公式 p = (n a + m b) / (m + n) 求得。务必练习把文字条件转化为向量方程。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Mistakes often happen when candidates forget that vectors must show both magnitude and direction. Never write a vector as just a number. Also, avoid mixing column vectors and unit vectors in a single expression without clear conversion. In collinearity proofs, simply stating that two vectors are scalar multiples is not enough: you must state that they share a common point. When calculating magnitude, double-check your squares and square roots; a sign error in components leads to an incorrect length. In the exam, use a ruler to draw vector arrows accurately if a diagram is required. Write all steps clearly, even if a question seems simple – WJEC awards method marks for correct vector routes. Finally, practice past paper questions under timed conditions so that you can handle the algebra confidently.

常见错误包括：忘记向量必须同时体现大小和方向，绝不能只用数字表示向量。同时，不要在同一个式子中混合使用列向量和单位向量而未做明确转换。在证明共线时，仅指出两个向量是标量倍数是不够的——你必须说明它们有公共点。计算模长时，仔细检查平方和开方；分量的符号错误会导致长度出错。考试中如果需要画图，用直尺准确绘制向量箭头。即使题目看起来简单，也要清晰写出所有步骤——WJEC 对正确的向量路径会给予方法分。最后，在计时条件下练习往年真题，确保你能自信地处理向量代数。

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WJEC IGCSE Maths: Vectors Revision Guide | IGCSE WJEC 数学：向量 考点精讲