📚 Vectors: Position, Lines and Applications | 向量：位置、直线与应用
Vectors are essential tools in A-Level Mathematics for describing quantities that have both magnitude and direction. In Edexcel papers, vector questions often combine position, displacement, scalar product and straight-line geometry in two or three dimensions.
向量是 A-Level 数学中描述既有大小又有方向的量的重要工具。在 Edexcel 试卷中，向量题常结合位置、位移、数量积以及二维或三维直线几何。
1. Vector Basics and Notation | 向量基础与表示法
In Edexcel A-Level Mathematics, a vector can be written in column form, such as a = (3, −1) in 2D or b = (2, 0, −4) in 3D. The standard basis vectors are i, j and k, so (3, −1) can be expressed as 3i − j.
在 Edexcel A-Level 数学中，向量可以写成列向量形式，如二维 a = (3, −1) 或三维 b = (2, 0, −4)。标准基向量为 i、j、k，因此 (3, −1) 可表示为 3i − j。
Adding vectors is done component-wise, and multiplying by a scalar stretches or shrinks the vector. Negative scalar multiplication reverses the direction.
向量按分量相加，乘以标量会拉伸或缩短向量。乘以负数标量则使方向反向。
(a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂ + b₂)
2. Magnitude and Unit Vectors | 模长与单位向量
The magnitude of a vector a = (x, y, z) is |a| = √(x² + y² + z²). A unit vector has magnitude 1 and is found by dividing a vector by its magnitude.
向量 a = (x, y, z) 的模长为 |a| = √(x² + y² + z²)。单位向量的模长为 1，求法是用向量的模去除该向量。
For example, if a = (1, 2, −2), then |a| = √(1² + 2² + (−2)²) = √9 = 3, so a unit vector in the direction of a is (1/3, 2/3, −2/3).
例如，若 a = (1, 2, −2)，则 |a| = √(1² + 2² + (−2)²) = √9 = 3，因此沿 a 方向的单位向量为 (1/3, 2/3, −2/3)。
3. Position Vectors and Displacement | 位置向量与位移
A position vector r = (x, y, z) gives the location of a point P relative to a fixed origin O. If A has position vector a and B has position vector b, then the displacement from A to B is AB = b − a.
位置向量 r = (x, y, z) 表示点 P 相对固定原点 O 的位置。若 A 的位置向量为 a，B 的位置向量为 b，则从 A 到 B 的位移为 AB = b − a。
This is crucial when finding distances between points, midpoints, or proving that three points are collinear.
在求两点间距离、中点或证明三点共线时，这一关系至关重要。
AB = b − a
4. Scalar (Dot) Product | 数量积（点积）
The scalar product of two vectors a and b is a·b = a₁b₁ + a₂b₂ + a₃b₃. It is called scalar because the result is a number, not a vector.
两个向量 a 和 b 的数量积为 a·b = a₁b₁ + a₂b₂ + a₃b₃。它被称为数量积，因为结果是标量而不是向量。
The dot product is also given by a·b = |a||b| cos θ, where θ is the angle between the vectors when they are placed tail to tail.
数量积也可表示为 a·b = |a||b| cos θ，其中 θ 是两向量尾对尾放置时的夹角。
a·b = |a||b| cos θ
5. Angle Between Vectors and Perpendicular Condition | 向量夹角与垂直条件
Rearranging the formula gives cos θ = (a·b) / (|a||b|). This is used to calculate the angle between two vectors exactly or to the nearest degree.
变形公式得到 cos θ = (a·b) / (|a||b|)，用于精确计算或近似计算两向量夹角。
If a and b are perpendicular, then θ = 90° so cos θ = 0, hence a·b = 0. Conversely, for non-zero vectors, a dot product of zero proves perpendicularity.
如果 a 与 b 垂直，则 θ = 90°，cos θ = 0，因此 a·b = 0。反过来，若向量非零且点积为零，则可证明垂直。
For parallel vectors, the dot product equals ±|a||b| depending on whether they point in the same or opposite direction.
对于平行向量，点积等于 ±|a||b|，正负取决于方向相同还是相反。
| Relationship | Condition | 中文 |
|---|---|---|
| Perpendicular | a·b = 0 | 垂直：点积为零 |
| Parallel | b = ka for scalar k | 平行：一向量是另一向量的标量倍数 |
6. Vector Equation of a Line | 直线的向量方程
A straight line can be written as r = a + λ b, where a is the position vector of one point on the line, b is a direction vector, and λ is a scalar parameter.
直线可写为 r = a + λ b，其中 a 是直线上某一点的位置向量，b 是方向向量，λ 是标量参数。
Every value of λ corresponds to a unique point on the line, so the same line can have many equivalent vector equations with different starting points or parallel direction vectors.
每个 λ 值对应直线上唯一一点，因此同一直线可以有多个等价向量方程，只要起点不同或方向向量平行即可。
r = a + λ b
7. Converting Between Vector and Cartesian Forms | 向量形式与笛卡尔形式的转换
In 3D, if r = (x, y, z), a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then the vector equation gives x = a₁ + λ b₁, y = a₂ + λ b₂, z = a₃ + λ b₃.
在三维中，若 r = (x, y, z)，a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，则向量方程给出 x = a₁ + λ b₁, y = a₂ + λ b₂, z = a₃ + λ b₃。
Eliminating λ leads to the Cartesian form (x − a₁)/b₁ = (y − a₂)/b₂ = (z − a₃)/b₃, provided no direction component is zero. If a component is zero, the corresponding coordinate is constant.
消去 λ 得到笛卡尔形式 (x − a₁)/b₁ = (y − a₂)/b₂ = (z − a₃)/b₃，前提是方向分量不为零。若某分量为零，对应坐标为常数。
(x − a₁)/b₁ = (y − a₂)/b₂ = (z − a₃)/b₃
8. Parallel, Intersecting and Skew Lines | 平行、相交与异面直线
Two lines are parallel if their direction vectors are multiples of each other. If the direction vectors are not parallel, the lines may intersect or be skew.
如果两条直线的方向向量互为倍数，则它们平行。若方向向量不平行，两直线可能相交或为异面直线。
To find an intersection, equate the two vector equations and solve for the parameters λ and μ. Then substitute back to check that all coordinates agree; if they do not, the lines are skew in 3D.
求交点时，令两个向量方程相等并解参数 λ 和 μ。然后代回检验所有坐标是否一致；若不一致，在三维中两直线为异面直线。
In 2D, non-parallel lines always intersect, but in 3D they can be skew.
在二维中，不平行的直线一定相交；但在三维中，不平行直线可能为异面直线。
9. Angle Between Two Lines | 两条直线之间的夹角
The angle between two straight lines is defined as the acute angle between their direction vectors. Use the direction vectors b and d in cos θ = |b·d| / (|b||d|).
两条直线之间的夹角定义为其方向向量之间的锐角。利用方向向量 b 和 d，cos θ = |b·d| / (|b||d|)。
The absolute value in the numerator ensures the acute angle is chosen, since reversing a direction vector gives the supplementary angle.
分子中的绝对值确保取锐角，因为方向向量反向会得到补角。
cos θ = |b·d| / (|b||d|)
10. Worked Problem-Solving Strategy | 解题策略实例
When tackling an Edexcel vector question, start by identifying known position vectors and direction vectors. Draw a diagram for geometric relationships, then translate statements such as ‘perpendicular’ or ‘parallel’ into dot product or scalar multiple conditions.
解答 Edexcel 向量题时，先确定已知位置向量和方向向量。画出几何关系示意图，然后将“垂直”或“平行”等语句转化为点积或标量倍数条件。
Always check your answers by substituting parameter values into original equations, especially when finding intersections or proving collinearity.
务必通过将参数值代回原方程来检验答案，尤其是求交点或证明共线时。
Common exam mistakes include forgetting the zero dot product for perpendicular vectors, using non-matching units,
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