📚 Differentiating Vectors | 向量微分
In A-Level Mechanics, many physical quantities such as displacement, velocity and acceleration are vectors that depend on time. To analyse motion in two or three dimensions, we need to differentiate vector functions with respect to time. This article explains the rules and applications of differentiating vectors, focusing on the Edexcel Mathematics specification.
在A-Level力学中，位移、速度和加速度等许多物理量都是依赖于时间的向量。为了分析二维或三维运动，我们需要对向量函数关于时间求导。本文讲解向量微分的规则和应用，紧扣Edexcel数学考试大纲。
1. Understanding Vector Functions | 理解向量函数
= x(t) i + y(t) j, where x(t) and y(t) are scalar functions representing the coordinates.
向量函数为每个变量（通常是时间 t）分配一个向量。用分量形式表示为 r(t) = x(t) i + y(t) j，其中 x(t) 和 y(t) 是表示坐标的标量函数。
motion is in three dimensions, we include a k component: r(t) = x(t) i + y(t) j + z(t) k.
如果运动是三维的，还需包含 k 分量：r(t) = x(t) i + y(t) j + z(t) k。
2. Component-wise Differentiation | 分量微分
separately. If r(t) = x(t) i + y(t) j, then the derivative dr/dt = (dx/dt) i + (dy/dt) j.
微分向量函数，只需分别对每个分量求导。若 r(t) = x(t) i + y(t) j，则导数 dr/dt = (dx/dt) i + (dy/dt) j。
This follows from the definition of the derivative as a limit, applied component by component.
这源于导数作为极限的定义，并对每个分量应用。
3. Rules of Vector Differentiation | 向量微分法则
The derivative of a constant vector a is zero: d/dt (a) = 0.
常数向量 a 的导数为零：d/dt (a) = 0。
derivative of a scalar multiple: d/dt (c u(t)) = c * du/dt, where c is a constant scalar.
数乘的导数：d/dt (c u(t)) = c * du/dt，其中 c 为常数标量。
Sum/difference rule: d/dt (u(t) ± v(t)) = du/dt ± dv/dt.
和差法则：d/dt (u(t) ± v(t)) = du/dt ± dv/dt。
Dot and cross product rules exist but are not required for Edexcel A-Level Maths (they are covered in Further Maths).
点积和叉积的求导法则在Edexcel A-Level数学中不作要求（属于进阶数学内容）。
4. Kinematic Quantities: Displacement, Velocity, Acceleration | 运动学量：位移、速度、加速度
r r(t) gives the displacement. Velocity v(t) = dr/dt. Acceleration a(t) = dv/dt = d²r/dt².
在运动学中，位置向量 r(t) 表示位移。速度 v(t) = dr/dt。加速度 a(t) = dv/dt = d²r/dt²。
These three vectors are linked by differentiation with respect to time.
这三个向量通过对时间求导相互关联。
5. Worked Example: Finding Velocity and Acceleration | 例题：求速度与加速度
Given r(t) = (3t² + 2t) i + (t³ − 5t) j, find velocity and acceleration vectors at time t.
已知 r(t) = (3t² + 2t) i + (t³ − 5t) j，求在时刻 t 的速度和加速度向量。
Differentiate: v(t) = dr/dt = (6t + 2) i + (3t² − 5) j.
求导：v(t) = dr/dt = (6t + 2) i + (3t² − 5) j。
Then a(t) = dv/dt = 6 i + (6t) j.
然后 a(t) = dv/dt = 6 i + (6t) j。
At t = 1, v = 8 i − 2 j and a = 6 i + 6 j.
当 t = 1 时，v = 8 i − 2 j，a = 6 i + 6 j。
6. Magnitude and Direction of Velocity | 速度的大小和方向
Speed is the magnitude of velocity: |v| = √vx² + vy². The direction is given by the angle θ with the positive i direction, where tanθ = vy / vx. Be careful with the quadrant.
速率是速度的大小：|v| = √vx² + vy²。方向由与正 i 方向的夹角 θ 给出，满足 tanθ = vy / vx。注意象限问题。
For example, if v = 3 i − 4 j, speed = 5, and direction is about −53.1° (or 306.9°).
例如，若 v = 3 i − 4 j，速率为 5，方向约为 −53.1°（或 306.9°）。
7. Second Derivative and Acceleration Vector | 二阶导数与加速度向量
Acceleration is the second derivative of the position vector. It indicates how velocity changes in both magnitude and direction.
加速度是位置向量的二阶导数。它表明速度的大小和方向如何变化。
otion is under uniform acceleration. In projectile motion, a = −g j (assuming g downward).
如果 a(t) 恒定，则为匀加速运动。在抛体运动中，a = −g j（假设 g 向下）。
8. Reversing the Process: Integrating Vectors | 逆过程：向量积分
time. v(t) = ∫ a(t) dt + C, where C is a constant vector determined by initial conditions.
要从加速度得到速度，需对各分量关于时间积分。v(t) = ∫ a(t) dt + C，其中 C 是由初始条件确定的常数向量。
essential for solving kinematics problems when acceleration is given as a function of time.
类似地，r(t) = ∫ v(t) dt + D。当加速度作为时间的函数给出时，这是求解运动学问题的关键。
9. Applications in Mechanics Problems | 力学问题中的应用
Typical exam questions provide a position vector or acceleration vector and ask for speed, direction, time when velocity is parallel to a given vector, or when the particle is at rest.
典型考题给出位置向量或加速度向量，要求计算速率、方向、速度平行于某给定向量的时刻，或质点静止的时刻。
u may need to solve v(t) = 0 for t, or set v parallel to i by making the j component zero.
你可能需要求解 v(t) = 0 得到 t，或者令 j 分量为零以使 v 平行于 i。
10. Connecting with Parametric Equations | 与参数方程的联系
ive dr/dt gives dx/dt and dy/dt, which are the horizontal and vertical velocity components.
位置向量 r(t) = x(t) i + y(t) j 等价于参数方程 x = x(t), y = y(t)。导数 dr/dt 给出了 dx/dt 和 dy/dt，即水平和竖直速度分量。
This idea links vector differentiation with parametric differentiation (Core Pure/Pure).
这一思想将向量微分与参数微分（纯数/核心数学）联系起来。
11. Common Pitfalls and Exam Tips | 常见错误与考试技巧
1. Forgetting to differentiate constant vectors: a constant added to r(t) disappears upon differentiation.
1. 忘记常数向量求导：加在 r(t) 中的常数在求导后消失。
2. Mixing units or components: always keep i and j components separate.
2. 单位或分量混淆：始终将 i 与 j 分量分开处理。
3. Signs: careful when substituting negative values.
3. 符号：代入负值时需特别小心。
4. When finding direction, check quadrant: use arctan carefully.
4. 求方向时注意象限：谨慎使用反正切。
12. Summary and Practice | 总结与练习
Differentiating vectors reduces to differentiating scalar components. Master this technique and its application in kinematics to excel in Edexcel Mechanics. Practice past paper questions involving variable acceleration vectors.
向量微分归结为对标量分量求导。掌握此技巧及其在运动学中的应用，方能在Edexcel力学中取得优异成绩。多练习涉及变加速度向量的历年考题。
Remember: r → v → a (differentiate), a → v → r (integrate).
记住：r → v → a（求导），a → v → r（积分）。
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