📚 Position Vectors and Vector Equations | 位置向量与向量方程
In IB Mathematics, vectors provide a powerful language for describing positions, displacements, and geometric figures in both two and three dimensions. Understanding position vectors and the vector equations of lines and planes is essential for solving geometry problems analytically. This revision article covers key definitions, techniques for constructing and interpreting vector equations, and methods for intersections, angles, and distances – all with clear examples and paired explanations.
在IB数学中，向量为描述二维和三维空间中的位置、位移和几何图形提供了强大的语言。理解位置向量以及直线和平面的向量方程，对于用解析方法解决几何问题至关重要。本篇复习文章涵盖关键定义、构建与解读向量方程的技巧，以及求交点、夹角和距离的方法，均配有清晰的示例和双语对照讲解。
1. What is a Position Vector? | 什么是位置向量？
A position vector is a vector that represents the location of a point relative to a fixed origin O. If a point P has coordinates (x, y, z), its position vector is denoted by r = OP. It can be written in component form as r = xi + yj + zk, where i, j, k are the standard base vectors.
位置向量表示一个点相对于固定原点O的位置。若点P的坐标为(x, y, z)，其位置向量记作 r = OP，可以写成分量形式 r = xi + yj + zk，其中 i, j, k 为标准基向量。
The position vector is always anchored at the origin; it identifies where the point is, not a direction of movement unless combined with another point.
位置向量始终以原点为起点；它指明点的位置，除非与另一个点结合使用，否则不代表运动方向。
2. Component Form of Vectors | 向量的分量形式
Any vector v can be expressed as a combination of the base vectors: v = vxi + vyj + vzk. For example, v = 3i - 2j + 4k has components (3, -2, 4). Its magnitude is |v| = √(3² + (-2)² + 4²) = √29.
任意向量 v 都可以用基向量的组合表示：v = vxi + vyj + vzk。例如，v = 3i - 2j + 4k 的分量为 (3, -2, 4)，其模为 |v| = √(3² + (-2)² + 4²) = √29。
When writing vectors in column form, the same vector is represented as a vertical array; this is often useful for organising systems of equations.
当用列向量表示时，同一向量写为垂直排列的数组；这在整理方程组时往往非常方便。
3. Free Vectors vs. Position Vectors | 自由向量与位置向量的区别
A free vector is defined only by its magnitude and direction; it does not have a fixed initial point. In contrast, a position vector always starts at the origin. For example, the displacement vector from point A to point B is AB = OB - OA, which is a free vector.
自由向量仅由其大小和方向定义，没有固定的起点。相反，位置向量总是从原点出发。例如，从点A到点B的位移向量 AB = OB - OA 就是一个自由向量。
This distinction is vital when working with vector equations: the a in r = a + λb is a position vector locating a point on the line, while the direction vector b is a free vector indicating the line's orientation.
在处理向量方程时，这一区别至关重要：方程 r = a + λb 中的 a 是确定直线上一点的位置向量，而方向向量 b 则是指示直线方向的自由向量。
4. Vector Equation of a Straight Line | 直线的向量方程
A line passing through a point with position vector a and parallel to a direction vector b is given by r = a + λb, where λ is a real parameter. Every value of λ corresponds to a unique point on the line.
一条经过位置向量为 a 的点且平行于方向向量 b 的直线可表示为 r = a + λb，其中 λ 为实数参数。每一个 λ 值对应直线上唯一的一点。
In 2D, if a = 2i + j and b = i - 3j, the line is r = (2i + j) + λ(i - 3j s become x = 2 + λ, y = 1 - 3λ, which can be converted to Cartesian form by eliminating λ.
在二维中，若 a = 2i + j，b = i - 3j，则直线为 r = (2i + j) + λ(i - 3j)。其参数方程为 x = 2 + λ，y = 1 - 3λ，消去 λ 即可转换为笛卡尔形式。
5. Finding the Vector Equation from Two Points | 由两点求向量方程
Given two points A and B with position vectors a and b, the direction vector for the line through A and B is d = b - a. The vector equation can be written as r = a + λ(b - a) or equivalently r = a + λd.
已知两点A和B，其位置向量分别为 a 和 b，则过A和B的直线的方向向量为 d = b - a。向量方程可写为 r = a + λ(b - a)，或等价地 r = a + λd。
This formulation is extremely practical when no explicit direction vector is given, but you know the coordinates of two points on the line.
当没有明确给出方向向量，但知道直线上两点的坐标时，这种方法极其实用。
6. Determining if a Point Lies on a Line | 判断点是否在直线上
To test whether a point C with position vector c lies on r = a + λb, set c = a + λb and solve for λ. If a single λ value satisfies all component equations, then C is on the line; if the equations are inconsistent, it is not.
要检验位置向量为 c 的点C是否在 r = a + λb 上，令 c = a + λb 并求解 λ。若存在一个 λ 值同时满足所有分量方程，则C在直线上；若方程组矛盾，则不在。
Example: a = i + 2k, b = 2i - j, c = 5i - 2j + 2k. From the i-component: 1 + 2λ = 5 → λ = 2. The j-component gives 0 - λ = -2 → λ = 2. The k-component gives 2 + 0λ = 2, which is always true. Hence the point lies on the line.
示例：a = i + 2k，b = 2i - j，
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