📚 Vectors | 向量
Vectors are fundamental mathematical objects used to represent quantities that have both magnitude and direction. In A‑Level Mathematics, vectors appear in pure mathematics, mechanics, and further mathematics, forming a bridge between algebra and geometry. This article covers the core Edexcel syllabus for vectors, including notation, operations, scalar product, and geometric applications.
向量是基本数学对象，用于表示既有大小又有方向的量。在A‑Level数学中，向量出现在纯数学、力学和进阶数学中，是代数与几何之间的桥梁。本文涵盖Edexcel考试大纲中向量的核心内容，包括符号表示、运算、数量积以及几何应用。
1. What is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. Examples include displacement, velocity, acceleration, and force. In contrast, a scalar has only magnitude, such as mass, speed, or distance. Vectors are often represented diagrammatically by arrows, where the length of the arrow indicates the magnitude and the arrowhead shows the direction.
向量是既有大小又有方向的量。例如位移、速度、加速度和力。相比之下，标量只有大小，如质量、速率或距离。向量通常用箭头图示表示，箭头长度表示大小，箭头指向表示方向。
In print, vectors are written in bold, e.g. v, or with an underline when handwritten, while in component form they are displayed vertically or horizontally as column or row vectors.
在印刷体中，向量用粗体表示，例如v，手写时加下划线，而在分量形式中，向量以列向量或行向量的形式竖直或水平书写。
2. Vector Notation and Component Form | 向量的表示法与分量形式
A two‑dimensional vector can be expressed as a column vector v = (x, y) or as a combination of the standard basis vectors i and j: v = xi + yj. In three dimensions, v = xi + yj + zk. The numbers x, y, z are called the components of the vector.
二维向量可以表示为列向量v = (x, y)，或用标准基向量i和j的组合表示：v = xi + yj。在三维空间中，v = xi + yj + zk。数字x、y、z称为向量的分量。
For example, a vector from point A(1,2) to point B(4,6) is given by AB = (4−1)i + (6−2)j = 3i + 4j. This is also written as the column vector (3, 4)ᵀ, where the superscript T denotes transpose for a row.
例如，从点A(1,2)到点B(4,6)的向量为AB = (4−1)i + (6−2)j = 3i + 4j。这也写成列向量(3, 4)ᵀ，其中上标T表示行向量的转置。
3. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector v = xi + yj is given by |v| = √x² + y². For a three‑dimensional vector xi + yj + zk, the magnitude is |v| = √x² + y² + z². The magnitude is always a non‑negative scalar.
向量v = xi + yj的模（长度）由|v| = √x² + y²给出。对于三维向量xi + yj + zk，模为|v| = √x² + y² + z²。模始终为非负标量。
A unit vector is a vector with magnitude 1. To find the unit vector in the direction of v, divide the vector by its magnitude: û = v / |v|.
单位向量是模为1的向量。要找到v方向上的单位向量，用向量除以其模：û = v / |v|。
For example, if v = 3i + 4j, then |v| = √3² + 4² = 5, and the unit vector is (3/5)i + (4/5)j.
例如，若v = 3i + 4j，则|v| = √3² + 4² = 5，单位向量为(3/5)i + (4/5)j。
4. Addition and Subtraction of Vectors | 向量的加法与减法
Vectors are added by adding their corresponding components. Geometrically, addition obeys the triangle or parallelogram law. To subtract a vector, add its negative: a − b = a + (−b).
向量相加时，将对应的分量分别相加。从几何上看，加法遵循三角形法则或平行四边形法则。向量减法即加上负向量：a − b = a + (−b)。
If a = 2i + 3j and b = −i + 5j, then a + b = (2−1)i + (3+5)j = i + 8j, and a − b = 3i − 2j.
若a = 2i + 3j，b = −i + 5j，则a + b = (2−1)i + (3+5)j = i + 8j，且a − b = 3i − 2j。
5. Scalar Multiplication and Parallel Vectors | 标量乘法与平行向量
Multiplying a vector by a scalar k stretches or shrinks its magnitude by a factor of |k|, and reverses direction if k is negative. The resulting vector is parallel to the original. Two vectors a and b are parallel if a = λb for some scalar λ.
向量乘以标量k会使其模伸缩|k|倍，若k为负则反向。所得向量与原向量平行。如果存在标量λ使得a = λb，则两向量a和b平行。
For instance, 3(2i − j) = 6i − 3j. This vector is parallel to 2i − j because one is a scalar multiple of the other.
例如，3(2i − j) = 6i − 3j。该向量与2i − j平行，因为一个是另一个的标量倍。
6. Position Vectors and Displacement | 位置向量与位移
The position vector of a point A is the vector from the origin O to A, often written as OA or simply a. For two points A and B with position vectors a and b, the displacement vector from A to B is AB = b − a.
点A的位置向量是从原点O到A的向量，常写作OA或简写作a。对于位置向量分别为a和b的两点A和B，从A到B的位移向量为AB = b − a。
Midpoints and ratios can be expressed using vector addition. The midpoint M of AB has position vector m = ( a + b ) / 2. A point dividing AB in the ratio m:n has position vector (na + mb) / (m + n).
中点和比值可以用向量加法表示。线段AB的中点M的位置向量为m = ( a + b ) / 2。按比例m:n分割AB的点的位置向量为(na + mb) / (m + n)。
7. The Scalar (Dot) Product | 数量积（点积）
The scalar product of two vectors a and b is defined as a · b = |a| |b| cos θ, where θ is the angle between them. In component form, for a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the dot product is a₁b₁ + a₂b₂ + a₃b₃.
两向量a和b的数量积定义为a · b = |a| |b| cos θ，其中θ是它们之间的夹角。在分量形式下，对于a = a₁i + a₂j + a₃k和b = b₁i + b₂j + b₃k，点积为a₁b₁ + a₂b₂ + a₃b₃。
The scalar product is commutative: a · b = b · a, and distributive over addition. It yields a scalar, not a vector.
数量积满足交换律：a · b = b · a，且对加法有分配律。其结果是一个标量，而非向量。
8. Angle Between Two Vectors | 两向量之间的夹角
From the definition, cos θ = (a · b) / (|a| |b|). This formula allows us to find the angle between two vectors when their components are known. If a · b = 0 and both vectors are non‑zero, then cos θ = 0, so θ = 90° and the vectors are perpendicular (orthogonal).
由定义，cos θ = (a · b) / (|a| |b|)。该公式可用于在已知分量时求两向量之间的夹角。如果a · b = 0 且两者均为非零向量，则 cos θ = 0，因此 θ = 90°，向量垂直（正交）。
Example: Find the angle between a = 3i + 4j and b = i − j. Dot product = 3×1 + 4×(−1) = −1. Magnitudes: |a| = 5, |b| = √2. Hence cos θ = −1 / (5√2), giving θ ≈ 98.1°.
示例：求a = 3i + 4j与b = i − j之间的夹角。点积 = 3×1 + 4×(−1) = −1。模：|a| = 5，|b| = √2。因此 cos θ = −1 / (5√2)，得出 θ ≈ 98.1°。
9. Vector Equations of Lines | 直线的向量方程
A straight line in 2D or 3D can be described by a vector equation: r = a + λd, where a is the position vector of a point on the line, d is a direction vector parallel to the line, and λ is a scalar parameter. As λ varies, r traces all points on the line.
二维或三维空间中的直线可以用向量方程描述：r = a + λd，其中a是直线上一点的位置向量，d是平行于直线的方向向量，λ是一个标量参数。随着λ变化，r描绘出直线上所有点。
For example, a line passing through (2,1) with direction 3i − 2j has equation r = (2i + j) + λ(3i − 2j). This can be written as parametric equations: x = 2 + 3λ, y = 1 − 2λ.
例如，一条经过点(2,1)且方向为3i − 2j的直线，其方程为r = (2i + j) + λ(3i − 2j)。这可以写成参数方程：x = 2 + 3λ，y = 1 − 2λ。
10. Intersection, Parallel and Skew Lines | 直线的相交、平行与异面
Two lines are parallel if their direction vectors are scalar multiples of each other. If they share a common point, they are coincident. For non‑parallel lines, solving simultaneous vector equations can determine if they intersect. In 3D, lines that are not parallel and do not intersect are called skew.
如果两条直线的方向向量互为标量倍数，则它们平行。若它们共用一点，则重合。对于不平行的直线，通过解联立向量方程可以判断它们是否相交。在三维空间中，既不平行也不相交的直线称为异面直线。
To find the intersection, set a + λd₁ = b + μd₂ and solve for λ and μ. If consistent solutions exist, the point of intersection can be found. If no solution exists and lines are not parallel, they are skew.
求交点时，令a + λd₁ = b + μd₂，并求解λ和μ。如果存在一致解，则可得出交点。若无解且直线不平行，则它们异面。
11. Application in Mechanics: Velocity and Force | 力学应用：速度与力
In mechanics, vectors are used to represent forces, velocities, accelerations, and displacements. Resultant force is the vector sum of all forces acting on a particle. Equilibrium occurs when the resultant force is the zero vector. Resolution of forces involves breaking a vector into perpendicular components, often along i and j directions.
在力学中，向量用于表示力、速度、加速度和位移。合力是作用在质点上的所有力的矢量和。当合力为零向量时，质点的合力平衡。力的分解涉及将向量分解为垂直分量，通常沿i和j方向。
For a particle moving with constant velocity v, its position vector at time t is r = r₀ + vt, where r₀ is the initial position. This is a direct application of the vector equation of a line.
对于以恒定速度v运动的质点，其在时刻t的位置向量为r = r₀ + vt，其中r₀为初始位置。这是直线向量方程的直接应用。
12. Key Formulae Summary | 关键公式总结
| Concept / 概念 | Formula / 公式 |
|---|---|
| Magnitude / 模 | |v| = √x² + y² [2D]; √x² + y² + z² [3D] |
| Unit vector / 单位向量 | û = v / |v| |
| Dot product / 点积 | a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃ |
| Angle between vectors / 向量夹角 | cos θ = (a · b) / (|a| |b|) |
| Vector equation of line / 直线向量方程 | r = a + λd |
| Midpoint / 中点 | m = (a + b) / 2 |
These formulas are essential for solving vector problems in coordinate geometry and mechanics. Mastering their application will give you confidence in both pure and applied contexts.
这些公式对于解决坐标几何和力学中的向量问题至关重要。掌握它们的应用将使你在纯数学和应用数学领域都充满信心。
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