Mixed Exercise 11: Vectors | 混合练习11：向量

📚 Mixed Exercise 11: Vectors | 混合练习11：向量

Mixed Exercise 11 in the Edexcel A Level Pure Mathematics course brings together the core vector techniques that frequently appear in AS and A Level exams. It is designed to test your ability to switch between column vectors and i,j notation, calculate magnitudes, work with position vectors, prove parallel or collinear properties, and solve geometric problems using vector algebra. Many students find the exercise challenging because it requires both algebraic accuracy and geometric interpretation.

Edexcel A Level 纯数学课程中的混合练习11汇集了AS和A Level考试中经常出现的核心向量技巧。该练习旨在检验你是否能够在列向量与 i,j 表示法之间切换、计算模长、处理位置向量、证明平行或共线性质，以及利用向量代数解决几何问题。许多学生觉得这个练习有难度，因为它同时要求代数准确性和几何直观。

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1. Overview of the Exercise | 练习概览

This mixed exercise consolidates the whole chapter on vectors from the Edexcel Pure Mathematics Year 1/AS textbook. Questions are usually grouped by skill, but later parts combine several ideas, such as finding a vector, then using it to prove collinearity or to locate a point on a line segment.

本混合练习整合了Edexcel纯数学第一年/AS教材中向量整章的内容。题目通常按技能分组，但后面的部分会综合多个知识点，例如先求向量，再利用它证明共线或确定线段上的点。

In the exam, vector questions are often worth 5 to 8 marks and appear on both the AS and A Level papers. The mixed exercise mirrors that style by including structured parts such as ‘find the vector AB’, ‘show that points are collinear’, and ‘find the value of k’.

在考试中，向量题通常占5到8分，出现在AS和A Level试卷中。混合练习模拟这种风格，包含结构化小题，例如“求向量AB”“证明三点共线”及“求k的值”。

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2. Vector Notation and Representation | 向量表示法

A vector can be written either as a column vector or in terms of the unit vectors i and j. For example, a vector with horizontal component 3 and vertical component −4 can be written as the column vector with entries 3 and −4, or as 3i − 4j.

向量可以写成列向量形式，也可以用单位向量 i 和 j 表示。例如，水平分量为3、垂直分量为−4的向量可以写成列向量（分量为3和−4）或 3i − 4j。

It is important to keep the order of components consistent: the first component is always the x-direction or i-direction, and the second component is the y-direction or j-direction. In column form this is written with x above y.

保持分量顺序一致很重要：第一个分量始终是 x 方向或 i 方向，第二个分量是 y 方向或 j 方向。在列向量形式中，x 写在 y 的上方。

a = xi + yj

a = (x, y)

Many mixed exercise questions begin by asking you to write down a vector in a particular form, so make sure you can convert freely between column notation and i,j notation.

许多混合练习题目一开始会要求你以某种特定形式写出向量，因此请确保你能在列向量表示法与 i,j 表示法之间自由转换。

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3. Magnitude and Unit Vectors | 模与单位向量

The magnitude of a vector a = xi + yj is given by |a| = √(x² + y²). This represents the length of the vector and is always non-negative. Magnitude questions often ask you to give an exact answer, so keep square roots simplified rather than rounding.

向量 a = xi + yj 的模由 |a| = √(x² + y²) 给出。它表示向量的长度，且始终为非负。模长题通常要求给出精确答案，因此请保留简化后的根式而不要取近似值。

|a| = √(x² + y²)

A unit vector has magnitude 1 and is found by dividing a vector by its magnitude: â = a / |a|. Unit vectors are useful for describing direction without changing magnitude.

单位向量的模为1，可由向量除以其模得到：â = a / |a|。单位向量用于描述方向而不改变大小。

In mixed exercise questions, you are often asked to find a unit vector in the direction of a given vector or to determine whether a vector is a unit vector. Always check that the magnitude of your final answer is actually 1.

在混合练习的题目中，经常要求你求给定向量方向上的单位向量，或判断某个向量是否为单位向量。请始终检查最终答案的模是否确实为1。

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

A position vector starts from the origin to a point P. If point A has coordinates (x₁, y₁), its position vector is a = x₁i + y₁j. This is usually written as OA.

位置向量从原点出发指向点 P。如果点 A 的坐标为 (x₁, y₁)，其位置向量为 a = x₁i + y₁j。通常写作 OA。

The displacement vector from point A to point B is found by subtracting the position vector of A from that of B: AB = b − a. This is one of the most frequently tested formulas in Mixed Exercise 11.

从点 A 到点 B 的位移向量由点 B 的位置向量减去点 A 的位置向量得到：AB = b − a。这是混合练习11中最常考查的公式之一。

AB = OB − OA = b − a

For example, if A is (2, 5) and B is (−1, 3), then AB = (−1 − 2)i + (3 − 5)j = −3i − 2j. Note that AB is not the same as BA; BA would be the opposite vector 3i + 2j.

例如，若 A 为 (2, 5)，B 为 (−1, 3)，则 AB = (−1 − 2)i + (3 − 5)j = −3i − 2j。注意 AB 与 BA 不同；BA 是相反向量 3i + 2j。

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5. Addition, Subtraction and Scalar Multiplication | 向量加减与数乘

Vectors can be added by combining their i-components and j-components separately. Subtraction works in the same way, component by component. For example, (3i + 2j) − (i − 5j) = 2i

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