📚 Vectors in Mechanics | 力学中的矢量
In A-level mechanics, a large number of quantities are vectors rather than scalars. Being able to add, resolve and interpret vectors is fundamental for forces, motion, moments and equilibrium problems in the Edexcel specification.
在 A-level 力学中，许多物理量是矢量而不是标量。能够正确地进行矢量的合成、分解与解读，是解决 Edexcel 大纲中力、运动、力矩和平衡问题的基础。
1. Scalars and Vectors | 标量与矢量
A scalar quantity has magnitude only. Mass, time, temperature, energy, distance and speed are scalars. A vector quantity has both magnitude and direction. Displacement, velocity, acceleration, force, weight, momentum and moment are all vectors.
标量只有大小。质量、时间、温度、能量、路程和速率都是标量。矢量既有大小又有方向。位移、速度、加速度、力、重力、动量和力矩都是矢量。
| Scalar 标量 | Vector 矢量 |
|---|---|
| distance 路程 | displacement 位移 |
| speed 速率 | velocity 速度 |
| mass 质量 | force 力 |
| energy 能量 | momentum 动量 |
2. Vector Representation and Notation | 矢量的表示与记号
A vector is drawn as an arrow. The length of the arrow is proportional to the magnitude of the vector, and the arrowhead points in the direction of the vector. In written material, a vector may be shown as a bold letter or with an arrow above the letter; its magnitude is written using the same letter without bold or between vertical bars.
矢量通常画成一个箭头。箭头的长度与矢量的大小成正比，箭头指向矢量的方向。在书面材料中，矢量可以用粗体字母或在字母上方加箭头表示；它的大小用不带粗体的同一字母或加绝对值符号表示。
For example, a force of 10 N acting at 30° above the horizontal can be written as F = 10 N at 30°. Its magnitude is F = |F| = 10 N.
例如，一个大小为 10 N、方向为水平线以上 30° 的力可以写作 F = 10 N，方向 30°。它的大小为 F = |F| = 10 N。
3. Adding and Subtracting Vectors | 矢量的加法与减法
Vectors are added tip-to-tail. The resultant vector is the single arrow drawn from the starting point to the final tip. If two perpendicular vectors A and B are added, the magnitude of the resultant is found using Pythagoras and the direction is found using trigonometry.
矢量按“首尾相接”的方法相加。合矢量是从起点画到最终末端的单一箭头。如果两个互相垂直的矢量 A 和 B 相加，合矢量的大小可用勾股定理求出，方向可用三角函数求出。
R = √(A² + B²)
θ = tan⁻¹(B / A)
To subtract a vector, add the negative of that vector. The negative vector has the same magnitude but the opposite direction, so A − B = A + (−B).
矢量相减时，可以加上该矢量的负矢量。负矢量的大小相同但方向相反，因此 A − B = A + (−B)。
4. Resolving Vectors into Components | 矢量分解为分量
Any vector can be resolved into two perpendicular components. This is especially useful in mechanics because horizontal and vertical motions, or parallel and perpendicular forces, can be treated independently. For a force F making an angle θ with the horizontal, the horizontal component is F cos θ and the vertical component is F sin θ.
任何矢量都可以分解为两个相互垂直的分量。这在力学中特别有用，因为水平和竖直运动，或平行与垂直的力，可以分别独立处理。对于与水平方向成 θ 角的力 F，水平分量为 F cos θ，竖直分量为 F sin θ。
Fₓ = F cos θ, F_y = F sin θ
If the angle is given to the vertical, the components are swapped: the component along the vertical is F cos θ, and the component along the horizontal is F sin θ.
如果角度是相对竖直方向给出的，则分量需要互换：沿竖直方向的分量为 F cos θ，沿水平方向的分量为 F sin θ。
5. Unit Vectors and Component Form | 单位矢量与分量形式
A unit vector has a magnitude of exactly 1 and is used to specify direction. In two-dimensional mechanics, the unit vector i points along the x-axis and the unit vector j points along the y-axis. A vector F can be written in component form as Fₓ i + F_y j.
单位矢量的大小恰好为 1，用来表示方向。在二维力学中，单位矢量 i 指向 x 轴正方向，单位矢量 j 指向 y 轴正方向。矢量 F 可以写成分量形式 Fₓ i + F_y j。
F = Fₓ i + F_y j
The magnitude of a vector expressed in component form is found using |F| = √(Fₓ² + F_y²). The direction angle relative to the x-axis is θ = tan⁻¹(F_y / Fₓ).
用分量形式表示的矢量，其大小可由 |F| = √(Fₓ² + F_y²) 求得。相对于 x 轴的方向角为 θ = tan⁻¹(F_y / Fₓ)。
6. Equilibrium and Resultant Force | 平衡与合力
An object is in equilibrium when the resultant force acting on it is zero. This means the vector sum of all forces is zero, so the forces form a closed vector polygon when drawn tip-to-tail. In component form, both the total horizontal component and the total vertical component must be zero.
当作用在物体上的合力为零时，物体处于平衡状态。这意味着所有力的矢量和为零，因此按首尾相接画出的力矢量形成一个闭合多边形。用分量形式表示时，总的水平分量和总的竖直分量都必须为零。
ΣFₓ = 0, ΣF_y = 0
For three non-parallel forces in equilibrium, the three forces must be concurrent and can be represented by the sides of a closed triangle. This is often useful in statics problems involving strings, rods and smooth surfaces.
对于三个互不平行的力处于平衡的情况，三个力必须共点，并且可以用一个闭合三角形的三条边来表示。这在涉及绳子、杆和光滑表面的静力学问题中非常有用。
7. Free-Body Diagrams and Vector Forces | 受力图与矢量力
A free-body diagram shows all the forces acting on a single body as vector arrows. Common forces include weight W = mg acting downwards, the normal reaction N perpendicular to a surface, tension T along a string or rod, and friction f opposing motion or potential motion.
受力图用一个物体上所有作用力的矢量箭头来表示。常见的力包括竖直向下的重力 W = mg、垂直于表面的法向反作用力 N、沿绳或杆方向的张力 T，以及阻碍运动或运动趋势的摩擦力 f。
For a block on a rough slope inclined at angle θ to the horizontal, the weight can be resolved into a component mg sin θ down the slope and a component mg cos θ perpendicular to the slope. If the block does not accelerate perpendicular to the slope, then the normal reaction equals mg cos θ.
对于放在与水平面成 θ 角的粗糙斜面上的物块，重力可以分解为沿斜面向下的分量 mg sin θ 和垂直于斜面的分量 mg cos θ。如果物块在垂直斜面方向没有加速度，则法向反作用力等于 mg cos θ。
W = mg, W_parallel = mg sin θ, W_perpendicular = mg cos θ
8. Motion Vectors: Displacement, Velocity and Acceleration | 运动矢量：位移、速度和加速度
Displacement, velocity and acceleration are vector quantities. In straight-line motion, a positive sign usually means one direction along the chosen axis and a negative sign means the opposite direction. The equations of uniform acceleration can be applied with these sign conventions.
位移、速度和加速度都是矢量。在直线运动中，正号通常表示沿选定坐标轴的一个方向，负号表示相反方向。匀加速运动方程可以在这种符号规定下使用。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
In projectile motion, the horizontal and vertical components of velocity are independent. The horizontal velocity is constant if air resistance is ignored, while the vertical acceleration is g = 9.81 m s⁻² downwards.
在抛体运动中，速度的水平分量和竖直分量相互独立。如果忽略空气阻力，水平速度保持不变，而竖直加速度为向下的 g = 9.81 m s⁻²。
9. Relative Velocity and Vector Applications | 相对速度与矢量应用
Relative velocity is the velocity of one object as seen by another. It is found by vector subtraction: the velocity of A relative to B is the velocity of A minus the velocity of B. This is important in problems involving aircraft in wind and boats crossing rivers.
相对速度是从另一个物体的参考系观察到的一个物体的速度。它通过矢量减法求得：A 相对于 B 的速度等于 A 的速度减去 B 的速度。这在涉及飞机遇风和小船过河的问题中非常重要。
v_AB = v_A − v_B
For example, if a boat heads due north at 4 m s⁻¹ across a river flowing east at 3 m s⁻¹, its resultant velocity relative to the bank is √(4² + 3²) = 5 m s⁻¹ at an angle tan⁻¹(3/4) east of north.
例如，一艘小船以 4 m s⁻¹ 的速度向正北方向过河，河水以 3 m s⁻¹ 的速度向东流动，则船相对河岸的合速度为 √(4² + 3²) = 5 m s⁻¹，方向为北偏东 tan⁻¹(3/4)。
10. Moments and Vector Direction | 力矩与矢量方向
A moment is a vector quantity that describes the turning effect of a force. Its magnitude is the product of the force and the perpendicular distance from the pivot to the line of action of the force. In two dimensions, the direction is either clockwise or anticlockwise.
力矩是描述力产生转动效果的矢量。它的大小等于力与从支点到力作用线的垂直距离的乘积。在二维问题中，力矩的方向为顺时针或逆时针。
M = F d
In three-dimensional vector notation, the moment of a force about a point is the cross product of the position vector and the force vector: M = r × F. This vector points along the axis of rotation according to the right-hand rule, which is why moment is treated as a vector quantity in advanced mechanics.
在三维矢量表示中，力对一点的力矩等于位置矢量与力矢量的叉积：M = r × F。该矢量按右手定则指向旋转轴方向，因此力矩在更高阶的力学中被视为矢量。
11. Common Mistakes and Exam Tips | 常见错误与考试提示
One common mistake is treating speed as a vector. Speed is a scalar; velocity is the vector. Another mistake is adding force magnitudes directly when the forces are not acting in the same straight line. Always draw a vector diagram or resolve into components before combining forces.
一个常见错误是把速率当作矢量。速率是标量，速度才是矢量。另一个错误是在力不在同一直线上时直接把力的数值相加。在合成力之前，一定要画矢量图或分解为分量。
When resolving vectors, check whether the angle is measured from the horizontal or the vertical. If it is measured from the vertical, the sine and cosine components are swapped relative to the usual horizontal-angle case. In equilibrium problems, always verify that the vector sum of forces is zero in two perpendicular directions, not just one.
分解矢量时，要注意角度是从水平方向还是竖直方向测量的。如果角度是从竖直方向测量的，正
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