AQA Mathematics: Vectors Key Points Revision | AQA 数学：向量考点精讲

📚 AQA Mathematics: Vectors Key Points Revision | AQA 数学：向量考点精讲

Vectors form a crucial part of the AQA A-level Mathematics specification, bridging the gap between pure algebra and geometry. They allow you to represent quantities with both magnitude and direction, solve geometrical problems efficiently, and understand straight-line equations in a new light. This revision guide breaks down every key topic you need to master – from basic notation to segment division and line equations – with clear examples and exam-focused insights.

向量是 AQA 数学 A-Level 考试中的核心内容，它连接了纯代数与几何。向量能够表示既有大小又有方向的量，帮助你高效解决几何问题，并从全新角度理解直线方程。本考点精讲将逐一拆解你需要掌握的所有核心主题——从基本表示法到线段分点和直线向量方程——配有清晰的示例与针对考试的要点解析。

1. Vectors: Basic Concepts and Notation | 向量基本概念与表示

A vector is a quantity that has both magnitude (size) and direction. In contrast, a scalar has only magnitude, such as mass or temperature. In the AQA course, vectors are usually written in bold type (v) or as column vectors, and can be expressed in two or three dimensions using the standard unit vectors i, j and k. For example, a = 3i − 2j or a = (x, y) as a column.

向量是既有大小又有方向的量。而标量只有大小，如质量或温度。在 AQA 课程中，向量通常用粗体（v）或列向量表示，并可用标准单位向量 i、j、k 在二维或三维中表达。例如 a = 3i − 2j 或写成列向量 (x, y)。

A vector can also be given by its magnitude and direction (as a bearing or angle from the horizontal). Converting between component form and magnitude-direction form is a skill frequently tested in AS questions. The component form is simply the horizontal and vertical displacements.

向量也可以由其大小和方向（以象限角或与水平线的夹角表示）给出。在分量形式和大小 – 方向形式之间进行转换是 AS 题目中常考的技能。分量形式就是水平和竖直位移。

2. Vector Addition and Scalar Multiplication | 向量的加法和标量乘法

Vectors can be added using the triangle law or parallelogram law. Algebraically, you simply add the corresponding components: if a = a₁i + a₂j and b = b₁i + b₂j, then a + b = (a₁ + b₁)i + (a₂ + b₂)j. Subtraction works similarly by reversing the direction of the second vector.

向量加法遵循三角形法则或平行四边形法则。在代数上，只需将对应的分量相加：若 a = a₁i + a₂j 且 b = b₁i + b₂j，则 a + b = (a₁ + b₁)i + (a₂ + b₂)j。减法类似，只需将第二个向量反向后再相加。

Multiplying a vector by a scalar (a number) changes its magnitude but not its direction (unless the scalar is negative, which reverses the direction). For a scalar k, ka = k(a₁i + a₂j) = (ka₁)i + (ka₂)j. If k > 1, the vector is stretched; if 0 < k < 1, it shrinks; if k < 0, the direction is reversed.

向量乘上一个标量（一个数）会改变其大小但不改变方向（若标量为负则会反方向）。对于标量 k，ka = k(a₁i + a₂j) = (ka₁)i + (ka₂)j。若 k > 1，向量伸长；若 0 < k < 1，向量缩短；若 k < 0，方向相反。

3. Magnitude of a Vector | 向量的模

The magnitude (or length) of a vector v = xi + yj in 2D is given by |v| = √(x² + y²). For a 3D vector v = xi + yj + zk, the magnitude is √(x² + y² + z²). This comes directly from Pythagoras’ theorem.

二维向量 v = xi + yj 的模（长度）由公式 |v| = √(x² + y²) 给出。对于三维向量 v = xi + yj + zk，其模为 √(x² + y² + z²)。这直接来自勾股定理。

Exam questions often ask you to find the distance between two points using the magnitude of the difference of their position vectors. For points A and B, the distance AB = |AB|, where AB = b − a (position vectors of B and A). This is essentially the length of the vector joining them.

考试题经常要求你利用位置向量差的模来求两点间的距离。对点 A 和 B，距离 AB = |AB|，其中 AB = b − a（B 和 A 的位置向量）。这本质上是连接两点的向量的长度。

4. Unit Vectors | 单位向量

A unit vector has a magnitude of 1 and is used purely to indicate direction. The standard basis vectors are i = (1,0), j = (0,1) in 2D, and i, j, k in 3D. To find a unit vector in the direction of any non-zero vector v, divide v by its own magnitude: û = v / |v|.

单位向量的模为 1，仅用于指示方向。标准的基向量在二维中为 i = (1,0)、j = (0,1)，三维中为 i、j、k。要找到任意非零向量 v 方向上的单位向量，只需将 v 除以其模：û = v / |v|。

You must be comfortable expressing a vector in both component form and in terms of its magnitude and direction. For instance, if a vector has magnitude 10 and acts at an angle of 30° to the horizontal, its horizontal component is 10 cos 30° and its vertical component is 10 sin 30°. This links vectors closely to trigonometry.

你必须熟练地用分量形式以及大小和方向来表达向量。例如，若一个向量大小为 10，且与水平方向夹角为 30°，则其水平分量为 10 cos 30°，竖直分量为 10 sin 30°。这把向量与三角学紧密联系起来。

5. Position Vectors and Distance Between Points | 位置向量与点间距

The position vector of a point P relative to the origin O is the vector OP, usually written as p. If P has coordinates (x, y), then p = xi + yj. The vector joining two points A (with position vector a) and B (with b) is AB = b − a. Notice the order: final − initial.

点 P 相对于原点 O 的位置向量是 OP，常写作 p。若 P 坐标为 (x, y)，则 p = xi + yj。连接两点 A（位置向量为 a）和 B（位置向量为 b）的向量为 AB = b − a。注意次序：终点 − 起点。

This simple subtraction is the foundation for nearly all vector geometry problems. The distance AB is just |AB|, and the midpoint M of AB has position vector m = (a + b)/2. These formulas work identically in three dimensions.

这个简单的减法是几乎全部向量几何问题的基础。距离 AB 就是 |AB|，而 AB 的中点 M 的位置向量为 m = (a + b)/2。这些公式在三维中同样适用。

6. Parallel Vectors | 平行向量

Two vectors a and b are parallel if one is a scalar multiple of the other, i.e., a = kb for some non-zero scalar k. If k > 0 they are in the same direction; if k < 0 they are opposite. Parallel vectors are at the heart of proving that lines are parallel or points are collinear.

若一个向量是另一个的标量倍，即 a = kb（k 为非零标量），则两向量平行。若 k > 0，它们同向；若 k < 0，则反向。平行向量是证明直线平行或点共线的核心。

To check whether two vectors are parallel, examine the ratios of their corresponding components. For example, (2, 4) and (3, 6) are parallel because 3/2 = 6/4 = 1.5. In exams you must write down the scalar k explicitly and show the relationship.

要判断两向量是否平行，可检查其对应分量的比。例如，(2, 4) 与 (3, 6) 平行，因为 3/2 = 6/4 = 1.5。在考试中你必须明确写出标量倍数 k 并展示出该关系。

7. Collinear Points | 共线点

Three points A, B, C are collinear if they lie on the same straight line. In vector terms, this means the vectors AB and AC (or BC) are parallel, and since they share a common point (A), the points must be collinear. More formally, AB = t AC for some scalar t.

若三点 A、B、C 在同一直线上，则它们共线。用向量来说，这意味着向量 AB 与 AC（或 BC）平行，且由于它们共享一个公共点 (A)，这三点必定共线。更形式化地说，存在标量 t 使 AB = t AC。

Typical exam question: “Given the position vectors of A, B and C, show that A, B and C are collinear.” The approach is to find two vectors connecting the points, say AB and BC, and show that AB = k BC. Working with position vectors simplifies the process immensely.

典型考题：“已知 A、B、C 的位置向量，证明 A、B、C 共线。” 解题思路是找到连接这些点的两个向量，如 AB 和 BC，然后证明 AB = k BC。利用位置向量能极大地简化过程。

8. Dividing a Line Segment (Ratio Theorem) | 线段的分点公式（定比分点）

If a point P divides the line segment AB in the ratio λ : μ (with AP : PB = λ : μ), then the position vector of P is given by p = (μa + λb) / (λ + μ). Note the crossover of λ and μ. This formula works for both internal and external division (using negative ratios if needed), but AQA mainly tests internal division.

若点 P 将线段 AB 按比例 λ : μ 分割（即 AP : PB = λ : μ），则 P 的位置向量为 p = (μa + λb) / (λ + μ)。注意 λ 与 μ 的交叉。该公式对内外分都适用（外分时用负比），但 AQA 主要考查内分。

You can derive this rather than memorise: P lies on AB, so AP = (λ/(λ+μ)) AB. Then p = a + AP = a + (λ/(λ+μ))(b − a) = (μa + λb)/(λ+μ). The derivation reinforces your understanding and is useful if the ratio is given in a different form.

你可以推导而非死记：P 在 AB 上，所以 AP = (λ/(λ+μ)) AB。那么 p = a + AP = a + (λ/(λ+μ))(b − a) = (μa + λb)/(λ+μ)。推导过程能加深理解，且当比例以不同形式给出时会很有用。

9. Vector Equation of a Line | 直线的向量方程

In AQA core mathematics, you must be able to use the vector equation of a straight line: r = a + td, where a is the position vector of a fixed point on the line, d is a direction vector parallel to the line, and t is a real parameter. As t varies, r generates all points on the line.

在 AQA 核心数学中，你必须会使用直线的向量方程：r = a + td，其中 a 是直线上某已知点的位置向量，d 是平行于直线的方向向量，t 为实数参数。当 t 变化时，r 生成直线上的所有点。

This form is incredibly useful for showing whether a point lies on a line, finding intersections, or determining if two lines are parallel. Two lines are parallel if their direction vectors are scalar multiples. Intersection problems involve setting a₁ + sd₁ = a₂ + td₂ and solving for s and t.

此形式在判断点是否在直线上、求交点或确定两线是否平行时极其有用。若两直线的方向向量成标量倍数，则它们平行。交点问题需要令 a₁ + sd₁ = a₂ + td₂ 并求解 s 和 t。

10. Geometric Applications and Proofs | 几何应用与证明

Vectors shine in geometric proofs, which are frequently tested in AQA exams. You might be asked to prove that a quadrilateral is a parallelogram, that a point is a midpoint, or that three points form a straight line. The key is to set up position vectors for the given points and then manipulate vector expressions to establish the required relationships.

向量在几何证明中表现出色，这也是 AQA 考试中常考的内容。你可能需要证明一个四边形是平行四边形、某点是中点，或者三点共线。关键是为已知点设出位置向量，然后对向量表达式进行运算，建立所需关系。

Common strategies: For a parallelogram ABCD, you must show AB = DC (or AD = BC). For a midpoint, demonstrate that m = (a + b)/2. For collinearity, as covered earlier, show AB = t AC. Always express vectors in terms of known position vectors and simplify.

常用策略：对于平行四边形 ABCD，须证明 AB = DC（或 AD = BC）。对于中点，要证 m = (a + b)/2。对于共线，如前所述，证明 AB = t AC。始终将向量用已知位置向量表示并化简。

When working in three dimensions, the same principles apply; just include the k component. Keep your working tidy and clearly state the vector equalities you are proving. Marks are awarded for logical structure as well as correct algebraic manipulation.

在三维空间中，原理相同，只需包含 k 分量。保持解答整洁，清晰陈述你所要证明的向量等式。得分点不仅在于代数运算正确，也在于逻辑结构清晰。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

AQA Mathematics: Vectors Key Points Revision | AQA 数学：向量 考点精讲