📚 Solving Geometry Problems with Vectors | A-Level 数学：用向量解几何问题
Vectors turn geometry into calculation. By representing points with position vectors and using a few algebraic rules, you can prove parallelism, perpendicularity, find intersections and compute distances without drawing complicated diagrams.
向量把几何问题转化为代数计算。通过用位置矢量表示点，再运用少量代数规则，你可以证明平行、垂直、求交点、算距离，而无需画复杂的图形。
1. Vector Notation and Position Vectors | 矢量记号与位置矢量
In A-Level mathematics a vector is a quantity with both magnitude and direction. You will often see vectors written in bold lower-case letters such as a or b, or in component form (x, y, z).
在 A-Level 数学中，矢量是既有大小又有方向的量。我们通常用粗体小写字母如 a、b 表示，或写成分量形式 (x, y, z)。
The unit vectors i, j and k point along the x-, y- and z-axes respectively. Any vector can be written as a combination of these unit vectors.
单位矢量 i、j、k 分别指向 x 轴、y 轴和 z 轴正方向。任何矢量都可以写成这些单位矢量的线性组合。
a = a₁i + a₂j + a₃k = (a₁, a₂, a₃)
A position vector describes the location of a point relative to the origin O. If point P has coordinates (x, y, z), its position vector is p = (x, y, z).
位置矢量描述点相对于原点 O 的位置。若点 P 的坐标为 (x, y, z)，则其位置矢量为 p = (x, y, z)。
2. Displacement Vectors and the Vector Triangle | 位移矢量与矢量三角形
Given two points A and B with position vectors a and b, the vector from A to B is a displacement vector.
已知两点 A、B 的位置矢量分别为 a、b，则由 A 指向 B 的矢量是一个位移矢量。
AB = b − a
Similarly, the vector from B to A is BA = a − b. Notice that AB and BA have the same magnitude but opposite directions.
类似地，从 B 指向 A 的矢量为 BA = a − b。注意 AB 与 BA 大小相同、方向相反。
Vector addition satisfies the triangle law: if AB is followed by BC, then the resultant is AC.
矢量加法满足三角形法则：先走 AB，再走 BC，合矢量就是 AC。
AC = AB + BC
This simple identity is the key to solving many geometric problems by vector algebra.
这一简单恒等式是用向量代数解决许多几何问题的关键。
3. Parallel Vectors, Scalar Multiples and Collinearity | 平行矢量、数乘与共线
Two non-zero vectors are parallel if one is a scalar multiple of the other. If
两个非零矢量平行，当且仅当其中一个可以写成另一个的实数倍。若
b = λa, λ ≠ 0
then a and b are parallel. If λ > 0, they point in the same direction; if λ < 0, they point in opposite directions.
则 a 与 b 平行。若 λ > 0，二者同向；若 λ < 0，二者反向。
Points A, B and C are collinear if the vectors AB and BC are parallel, or equivalently if there exists a scalar t such that
点 A、B、C 共线，当且仅当矢量 AB 与 BC 平行，也就是存在实数 t，使得
AB = tBC
This condition is used constantly in proofs of collinearity and in checking whether a point lies on a straight line.
这个条件经常用于证明三点共线，以及判断一个点是否在直线上。
4. Vector Equation of a Straight Line | 直线的矢量方程
A straight line can be described by one known position vector a and one direction vector d. The vector equation is
一条直线可以用一个已知点的位置矢量 a 和一个方向矢量 d 来描述。其矢量方程为
r = a + λd
where r is the position vector of a general point on the line and λ is a scalar parameter.
其中 r 是直线上任意一点的位置矢量，λ 是实参数。
For example, the line through A(1, 2) with direction vector (−3, 4) has equation
例如，过点 A(1, 2) 且方向矢量为 (−3, 4) 的直线方程为
r = (1, 2) + λ(−3, 4) = (1 − 3λ, 2 + 4λ)
In 3D, the vector equation has three component equations. To check whether a point lies on a line, substitute its coordinates and solve for the same value of λ in all three components.
在三维空间中，矢量方程对应三个分量方程。要判断某点是否在直线上，就把坐标代入，并检查能否在所有三个分量中解出同一个 λ 值。
5. The Dot Product and Angles Between Vectors | 点积与夹角
The dot product, or scalar product, of two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) can be written in two equivalent ways.
两个矢量 a = (a₁, a₂, a₃) 与 b = (b₁, b₂, b₃) 的点积（又称数量积）有两种等价写法。
a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ
where θ is the angle between the two vectors when they are placed tail to tail.
其中 θ 是两个矢量平移到“共用起点”后的夹角。
Rearranging gives the angle formula:
变形后得到夹角公式：
cosθ = (a · b) / (|a||b|)
This formula is essential for finding angles between lines, planes and geometric edges in vector geometry problems.
这个公式是求直线之间、平面之间以及几何棱边之间夹角的重要工具。
6. Perpendicular and Parallel Conditions Revisited | 再谈垂直与平行的条件
The dot product gives a very quick test for perpendicular vectors. If a and b are non-zero, then
点积为判断垂直提供了非常快捷的办法。若 a、b 均为非零矢量，则
a · b = 0 ⇔ a ⊥ b
This is because cos90° = 0.
这是因为 cos90° = 0。
For parallel vectors, we can also use the dot product result
对于平行矢量，也可以利用点积结论
|a · b| = |a||b|
but the most direct test remains b = λa. In exam questions, always check whether you are asked to prove parallel or perpendicular, and choose the appropriate condition.
但最直接的判断仍然是 b = λa。在考试中，先看清题目要求证明平行还是垂直，再选择相应条件。
7. The Projection of One Vector onto Another | 一个矢量在另一个矢量上的投影
The projection of a onto b is the component of a in the direction of b. Its scalar length is
a 在 b 方向上的投影，就是 a 沿 b 方向的分量。其投影长度为
scalar projection = (a · b) / |b|
The corresponding vector projection is obtained by multiplying this scalar by the unit vector in the direction of b.
相应的投影矢量，就是把上面的投影长度乘以 b 方向上的单位矢量。
vector projection = ( (a · b) / |b|² ) b
Projections are particularly useful for resolving forces and for computing the perpendicular distance from a point to a line.
投影特别适合用于力的分解，也用于计算点到直线的垂直距离。
8. Intersections of Lines in 2D and 3D | 二维与三维空间中直线的交点
To find the intersection of two lines given as
要求两条直线的交点，已知它们的方程为
r = a + λd and r = c + μe
equate the two expressions for r and solve the simultaneous component equations.
令两个 r 的表达式相等，然后解分量方程组。
For example:
例如：
