A-Level Maths: Vector Operations and Applications in Mechanics | A-Level 数学：力学中的向量运算与应用

📚 A-Level Maths: Vector Operations and Applications in Mechanics | A-Level 数学：力学中的向量运算与应用

Vectors are fundamental for describing motion and forces in A-Level mechanics. They link geometry, algebra and physical intuition, allowing us to represent quantities that have both magnitude and direction.

在 A-Level 力学中，向量是描述运动与力的基础。它将几何、代数与物理直觉联系起来，使我们能够表示同时具有大小和方向的量。

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1. Vector Notation and Basic Definitions | 向量记号与基本定义

In A-Level mathematics, a vector is defined by its magnitude (length) and direction. In mechanics, vectors often appear as directed line segments or as components in a coordinate system.

在 A-Level 数学中，向量由大小（长度）和方向定义。在力学中，向量通常以有向线段或坐标系中的分量形式出现。

Two common notations are column vectors and unit vector form. For a vector with components 3 in the x-direction and 4 in the y-direction, we write:

两种常见的记号是列向量和单位向量形式。对于在 x 方向分量为 3、y 方向分量为 4 的向量，我们写为：

v = (3, 4) = 3i + 4j

where i and j are unit vectors along the x- and y-axes respectively. In three dimensions, we add the unit vector k along the z-axis.

其中 i 和 j 分别是沿 x 轴和 y 轴的单位向量。在三维空间中，我们加上沿 z 轴的单位向量 k。

The magnitude of v is found using Pythagoras’ theorem:

向量 v 的大小由勾股定理求得：

|v| = √(3² + 4²) = 5

The direction is often given as an angle θ from the positive x-axis, with tan θ = y/x.

方向通常以与 x 轴正方向的夹角 θ 表示，满足 tan θ = y/x。

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2. Vector Addition, Subtraction and Resultants | 向量加法、减法与合向量

When several forces or velocities act on a particle, we add their vectors to find the resultant. For vectors a = (a₁, a₂) and b = (b₁, b₂):

当一个质点受到多个力或多个速度作用时，我们将这些向量相加得到合向量。对于向量 a = (a₁, a₂) 与 b = (b₁, b₂)：

a + b = (a₁ + b₁, a₂ + b₂)

Subtraction gives the vector from one point to another, or the change in a quantity:

相减则得到从一点到另一点的向量，或某个量的变化量：

a − b = (a₁ − b₁, a₂ − b₂)

For example, if a boat has velocity v = (2, 3) m/s and the river current is u = (1, −1) m/s, the resultant velocity is v + u = (3, 2) m/s.

例如，若船的速度为 v = (2, 3) m/s，水流速度为 u = (1, −1) m/s，则合速度为 v + u = (3, 2) m/s。

Graphically, addition follows the triangle or parallelogram law. This is especially useful for force diagrams in equilibrium problems.

在图形上，向量加法遵循三角形法则或平行四边形法则。这在平衡问题的受力分析中尤其有用。

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3. The Scalar (Dot) Product | 数量积（点乘）

The scalar product of two vectors a and b is defined as:

两个向量 a 与 b 的数量积定义为：

a · b = |a| |b| cos θ

where θ is the angle between the vectors. In component form, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then:

其中 θ 是两个向量之间的夹角。在分量形式中，若 a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，则：

a · b = a₁b₁ + a₂b₂ + a₃b₃

This product is used to find angles, test perpendicularity (a · b = 0), and later to compute work done.

数量积用于求夹角、判断垂直（a · b = 0），以及在后续计算功。

Example: For a = 3i + 4j and b = 2i − j, the scalar product is 3×2 + 4×(−1) = 6 − 4 = 2.

例如：对于 a = 3i + 4j 和 b = 2i − j，数量积为 3×2 + 4×(−1) = 6 − 4 = 2。

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4. Position Vectors and Displacement | 位置向量与位移

A position vector gives the location of a particle relative to a fixed origin O. It is usually written as r = xi + yj + zk.

位置向量表示质点相对于固定原点 O 的位置，通常写为 r = xi + yj + zk。

If a particle moves from point A with position vector a to point B with position vector b, the displacement vector is:

若质点从位置向量为 a 的点 A 运动到位置向量为 b 的点 B，则位移向量为：

s = b − a

The distance between two points is the magnitude of this displacement: |s|.

两点之间的距离就是这个位移的大小：|s|。

In mechanics, knowing initial and final position vectors allows us to determine the average velocity vector and the direction of motion.

在力学中，知道初末位置向量就能求出平均速度向量和运动方向。

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5. Velocity and Acceleration Vectors | 速度向量与加速度向量

Velocity is the rate of change of displacement with respect to time. As a vector:

速度是位移对时间的变化率。作为向量：

v = dr/dt

Acceleration is the rate of change of velocity:

加速度是速度对时间的变化率：

a = dv/dt = d²r/dt²

If the velocity vector is v = vₓi + vᵧj, then speed is |v| = √(vₓ² + vᵧ²). The direction of motion is the direction of v.

若速度向量为 v = vₓi + vᵧj，则速率为 |v| = √(vₓ² + vᵧ²)。运动方向就是 v 的方向。

For example, if r = (3t² + 2t)i + (5t − 1)j, then v = dr/dt = (6t + 2)i + 5j, and a = dv/dt = 6i.

例如，若 r = (3t² + 2t)i + (5t − 1)j，则 v = dr/dt = (6t + 2)i + 5j，a = dv/dt = 6i。

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6. Constant Acceleration Equations in Vector Form | 向量形式的匀加速运动方程

The constant acceleration (SUVAT) equations can be written vectorially. For initial velocity u, final velocity v, acceleration a, displacement s and time t:

匀加速直线运动（SUVAT）方程可以写成向量形式。对于初速度 u、末速度 v、加速度 a、位移 s 和时间 t：

v = u + at

s = ut + ½at²

v · v = u · u + 2a · s

Here, each quantity is a vector except t and the scalar product forms. These equations are particularly useful for projectile motion and problems with forces in two dimensions.

这里除 t 和数量积形式外，每个量都是向量。这些方程在抛体运动以及二维受力问题中非常有用。

Example: A particle starts at the origin with u = (2, 0) m/s and a = (0, −10) m/s². After 3 s, its velocity is v = (2, 0) + (0, −30) = (2, −30) m/s.

例：一质点从原点出发，u = (2, 0) m/s，a = (0, −10) m/s²。3 秒后，其速度为 v = (2, 0) + (0, −30) = (2, −30) m/s。

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7. Forces as Vectors and Newton’s Second Law | 力作为向量与牛顿第二定律

Force is a vector. Newton’s second law states:

力是向量。牛顿第二定律表明：

F = ma

where F, m and a are the net force, mass and acceleration vector. Since m is a scalar, F and a always point in the same direction.

其中 F、m 和 a 分别是合外力、质量和加速度向量。由于 m 是标量，F 与 a 方向始终相同。

In two dimensions, this gives two component equations:

在二维情形下，这给出两个分量方程：

Fₓ = maₓ, Fᵧ = maᵧ

For connected particles or inclined planes, we often resolve forces into components parallel and perpendicular to a direction before applying F = ma.

对于连接体或斜面问题，我们通常先将力沿某方向分解为平行和垂直分量，再应用 F = ma。

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8. Equilibrium of Particles | 质点的平衡

A particle is in equilibrium when the resultant force on

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