📚 Working with Vectors | 向量运算
Vectors are fundamental in A-Level Mathematics because they describe quantities that have both magnitude and direction. In the Edexcel specification, working with vectors includes representation, arithmetic, scalar product, and geometric applications in two and three dimensions.
向量在 A-Level 数学中十分基础，因为它们描述既有大小又有方向的量。在 Edexcel 考纲中，向量运算包括向量的表示、加减数乘、标量积以及二维和三维中的几何应用。
1. Vector Notation and Representation | 向量表示法
A vector can be written as a bold letter such as a, or as a directed line segment AB from point A to point B. Vectors differ from scalars such as distance or speed because they require both magnitude and direction. In component form, a two-dimensional vector is written using basis vectors: a = xi + yj.
向量可以写成粗体字母，例如 a，或写成从点 A 到点 B 的有向线段 AB。向量不同于距离或速率等标量，因为它需要同时给出大小和方向。在分量形式中，二维向量使用基向量表示：a = xi + yj。
a = x i + y j = (x, y)
In three dimensions, the vector is written as a = xi + yj + zk, where i, j, and k are unit vectors along the x-, y-, and z-axes.
在三维空间中，向量写作 a = xi + yj + zk，其中 i、j、k 分别是沿 x 轴、y 轴和 z 轴方向的单位向量。
2. Magnitude and Direction | 模与方向
The magnitude of vector a = xi + yj is its length, found using Pythagoras’ theorem: |a| = √(x² + y²). For a three-dimensional vector, the magnitude is |a| = √(x² + y² + z²).
向量 a = xi + yj 的模就是它的长度，可用勾股定理求得：|a| = √(x² + y²)。对于三维向量，模为 |a| = √(x² + y² + z²)。
|a| = √(x² + y²)
The direction of a two-dimensional vector is usually given by the angle θ measured from the positive x-axis. The tangent of this angle is tan θ = y/x, but you must always check the quadrant, because the inverse tangent function only gives angles between −90° and 90°.
二维向量的方向通常用从 x 轴正方向量起的角 θ 来表示。这个角的正切为 tan θ = y/x，但你必须检查象限，因为反正切函数只给出 −90° 到 90° 之间的角。
For example, if a = 3i + 4j, then |a| = √(3² + 4²) = 5, and tan θ = 4/3, so θ ≈ 53.1°.
例如，若 a = 3i + 4j，则 |a| = √(3² + 4²) = 5，且 tan θ = 4/3，因此 θ ≈ 53.1°。
3. Position Vectors and Component Form | 位置向量与分量形式
The position vector of a point P relative to the origin O is OP = p. If P has coordinates (x, y, z), then p = xi + yj + zk. The vector from A to B is AB = b − a, where a and b are the position vectors of A and B.
点 P 相对于原点 O 的位置向量是 OP = p。如果 P 的坐标为 (x, y, z)，则 p = xi + yj + zk。从 A 到 B 的向量为 AB = b − a，其中 a 和 b 分别是 A 和 B 的位置向量。
AB = b − a
This formula is very useful because it allows you to find the vector between any two points when their coordinates are known. If A = (2, 5) and B = (7, 1), then AB = (7 − 2)i + (1 − 5)j = 5i − 4j.
这个公式非常有用，因为当已知两点坐标时，可以用它求出任意两点之间的向量。若 A = (2, 5)，B = (7, 1)，则 AB = (7 − 2)i + (1 − 5)j = 5i − 4j。
4. Vector Addition and Subtraction | 向量加减法
Vectors are added component-wise: if a = x₁i + y₁j and b = x₂i + y₂j, then a + b = (x₁ + x₂)i + (y₁ + y₂)j. Geometrically, addition follows the triangle or parallelogram law. Subtraction a − b is equivalent to adding the negative of b.
向量按分量相加：若 a = x₁i + y₁j，b = x₂i + y₂j，则 a + b = (x₁ + x₂)i + (y₁ + y₂)j。几何上，加法遵循三角形法则或平行四边形法则。减法 a − b 等于加上 b 的负向量。
a + b = (x₁ + x₂) i + (y₁ + y₂) j
For example, if a = 2i + 3j and b = −i + 4j, then a + b = i + 7j, and a − b = 3i − j. The negative of a vector has the same magnitude but opposite direction.
例如，若 a = 2i + 3j，b = −i + 4j，则 a + b = i + 7j，并且 a − b = 3i − j。一个向量的负向量与原向量大小相同，但方向相反。
5. Scalar Multiplication and Parallel Vectors | 数乘与平行向量
Multiplying a vector by a scalar k changes its magnitude by |k| and reverses direction if k is negative. Two vectors a and b are parallel if one is a scalar multiple of the other: b = ka. This condition is often used to prove that points are collinear.
向量乘以标量 k 会使其模变为原来的 |k| 倍，若 k 为负则方向相反。两个向量 a 与 b 平行，当且仅当一个是另一个的标量倍：b = ka。这一条件常用于证明点共线。
b = k a
For example, if a = 2i + 4j and b = i + 2j, then b = ½a, so the vectors are parallel. In geometry, if AB = kBC, then A, B, and C lie on the same straight line.
例如，若 a = 2i + 4j，b = i + 2j，则 b = ½a，因此这两个向量平行。在几何中，若 AB = kBC，则 A、B、C 三点在同一条直线上。
6. Unit Vectors and Basis Vectors | 单位向量与基向量
A unit vector has magnitude 1. The unit vector in the direction of a is found by dividing a by its magnitude: a / |a|. The standard basis vectors are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1) in three dimensions.
单位向量的模为 1。与向量 a 同方向的单位向量可通过将 a 除以其模得到：a / |a|。标准基向量在三维中为 i = (1, 0, 0)、j = (0, 1, 0)、k = (0, 0, 1)。
unit vector = a / |a|
For example, if a = 3i + 4j, then |a| = 5, so the unit vector in the direction of a is (3/5)i + (4/5)j. Unit vectors are particularly useful when describing direction without changing magnitude.
例如，若 a = 3i + 4j，则 |a| = 5，因此与 a 同方向的单位向量为 (3/5)i + (4/5)j。单位向量在只描述方向而不改变大小时特别有用。
7. The Scalar (Dot) Product | 标量积（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 the vectors. In component form, for a = x₁Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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