📚 Modelling with Vectors | 向量建模
In A-Level Mathematics, vectors extend beyond pure geometry to model real-world situations, particularly in mechanics. From describing the motion of a particle to analysing forces acting on an object, vectors provide a concise and powerful language for modelling in two and three dimensions. This article explores the key techniques of modelling with vectors, linking algebraic vector operations to physical concepts such as displacement, velocity, acceleration, and force.
在A-Level数学中，向量超越了纯几何，可用来对现实世界的场景进行建模，尤其适用于力学。从描述粒子的运动，到分析作用在物体上的力，向量为二维和三维建模提供了一种简洁而强大的语言。本文将探讨向量建模的核心技巧，将向量代数运算与位移、速度、加速度和力等物理概念联系起来。
1. Introduction to Vector Modelling | 向量建模导论
Vectors are quantities that have both magnitude and direction, making them ideal for representing physical quantities like displacement, velocity, and force. In modelling, we often use unit vectors i and j (and k in 3D) to break down these quantities into components. This allows us to apply algebraic methods to solve mechanical problems.
向量是既有大小又有方向的量，因此非常适合表示位移、速度和力等物理量。在建模中，我们常使用单位向量 i 和 j（三维中加入 k）将这些量分解为分量，从而能够运用代数方法解决力学问题。
Modelling with vectors means translating a physical scenario into vector equations, solving them using vector algebra, and then interpreting the results back in the physical context. For example, a ship moving with a given velocity relative to water and a current can be modelled by adding the velocity vectors.
向量建模意味着将物理情景转化为向量方程，使用向量代数求解，再将结果解读回物理情景。例如，一艘船以给定速度相对于水运动，同时受水流影响，就可以通过叠加速度向量来建模。
2. Position Vectors and Displacement | 位置向量与位移
A position vector locates a point relative to a fixed origin. If a particle is at point P with coordinates (x, y), its position vector is r = xi + yj. Displacement is the change in position: from point A to point B, the displacement vector is AB = rB − rA.
位置向量给出一点相对于固定原点的位置。若质点位于坐标 (x, y) 的点 P，其位置向量为 r = xi + yj。位移是位置的改变量：从点 A 到点 B 的位移向量为 AB = rB − rA。
In modelling motion, the displacement vector as a function of time t, r(t), describes the path of the particle. By expressing r(t) in terms of its i and j components, we can separately analyse motion in perpendicular directions.
在运动建模中，作为时间 t
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