Vector Calculations in Physics | 物理中的向量计算

📚 Vector Calculations in Physics | 物理中的向量计算

Physics describes the natural world through quantities that can be measured and related mathematically. Among these quantities, vectors play a fundamental role because they capture not just 'how much' but also 'which way'. Understanding how to manipulate vectors – adding, subtracting, resolving, multiplying – is essential for tackling topics from kinematics to electromagnetism. This article explores the core vector calculations needed for IB Physics, with clear explanations and worked examples.

物理学通过可测量并能用数学关联的量来描述自然界。在这些量中，向量起着基础作用，因为它们不仅描述了“有多少”，还表明了“朝哪个方向”。掌握如何操作向量——加法、减法、分解、乘法——对于解决从运动学到电磁学的课题至关重要。本文探讨 IB 物理所需的核心向量计算，并提供清晰的解释和计算实例。

1. What Are Vectors? | 什么是向量？

In physics, quantities are divided into scalars and vectors. A scalar is a quantity that has only magnitude (size), such as mass (5 kg), temperature (300 K) or time (10 s). A vector, however, possesses both magnitude and direction. Common vector quantities include displacement (5 m north), velocity (20 m s⁻¹ east), acceleration and force.

在物理学中，物理量分为标量和向量。标量只有大小（量值），例如质量（5 kg）、温度（300 K）或时间（10 s）。而向量既有大小又有方向。常见的向量包括位移（向北 5 m）、速度（向东 20 m s⁻¹）、加速度和力。

Vectors are often represented by an arrow: the length is proportional to the magnitude, and the arrowhead shows the direction. In written form, we use bold type or an arrow above the letter, such as v or v⃗. Throughout this article we adopt the arrow notation, e.g. v⃗, F⃗.

向量常用一个箭头表示：箭头的长度与大小成正比，箭头指向表示方向。在书写时，我们用粗体或在字母上方加箭头，例如 v 或 v⃗。本文统一采用箭头上标，如 v⃗、F⃗。

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2. Representing Vectors | 向量的表示

In two dimensions, any vector v⃗ can be described by its components along perpendicular axes, usually labelled x and y. If we know the vector's magnitude v and its angle θ relative to the positive x‑axis, the components are v_x = v cosθ and v_y = v sinθ. The vector is then written as v⃗ = (v_x, v_y) or, using unit vectors, v⃗ = v_x î + v_y ĵ.

在二维中，任意向量 v⃗ 可以用沿互相垂直的坐标轴（通常记为 x 和 y）的分量来描述。若已知向量的大小 v 及其与 x 轴正方向的夹角 θ，则分量为 v_x = v cosθ、v_y = v sinθ。该向量可写作 v⃗ = (v_x, v_y) 或利用单位向量表示为 v⃗ = v_x î + v_y ĵ。

Unit vectors î, ĵ (and k̂ in three dimensions) have a magnitude of exactly 1 and point along the coordinate axes. They allow us to treat vector equations as algebraic combinations of components.

单位向量 î、ĵ（三维中还有 k̂）的大小恰好为 1，并指向坐标轴方向。它们让我们能够将向量方程当作分量的代数组合来处理。

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3. Vector Addition and Subtraction | 向量的加法和减法

Two vectors can be added graphically using the triangle rule (tip‑to‑ the parallelogram rule. Algebraically, you simply add the corresponding components. For a⃗ = (a_x, a_y) and b⃗ = (b_x, b_y), the resultant c⃗ = a⃗ + b⃗ has components c_x = a_x + b_x and c_y = a_y + b_y.

两个向量可以用三角形法则（首尾相接）或平行四边形法则进行图解加法。在代数上，只需将对应的分量相加。对于 a⃗ = (a_x, a_y) 和 b⃗ = (b_x, b_y)，合向量 c⃗ = a⃗ + b⃗ 的分量为 c_x = a_x + b_x、c_y = a_y + b_y。

Subtraction is equivalent to adding the negative of a vector: a⃗ - b⃗ = a⃗ + (-b⃗ of a vector has the same magnitude but opposite direction, so its components change sign.

减法相当于加上一个向量的负向量：a⃗ - b⃗ = a⃗ + (-b⃗)。一个向量的负向量大小相同但方向相反，因此其分量变号。

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4. Resolving Vectors into Components | 向量的分解

Resolution is the process of breaking a single vector into two perpendicular components, usually horizontal and vertical. This is the reverse of vector addition. For a force F⃗ making an angle θ with the horizontal, the components are F_x = F cosθ and F_y = F sinθ.

分解是将一个向量拆分为两个互相垂直的分量（通常是水平和竖直分量）的过程，它是向量加法的逆过程。对于与水平方向成 θ 角的力 F⃗，其分量为 F_x = F cosθ、F_y = F sinθ。

Component	Expression
Adjacent to angle	Fx = F cosθ
Opposite to angle	Fy = F sinθ

Always define your angle relative to a known axis, and check that your components make sense for the physical situation.

务必明确你所用的角度是相对于哪条已知轴的，并检查分量在物理情境下是否合理。

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5. Magnitude and Direction | 大小和方向

If you know the components v_x and v_y, the magnitude of the vector is obtained using Pythagoras' theorem:

若已知分量 v_x 和 v_y，向量的大小可用勾股定理求得：

|v⃗| = √v_x² + v_y²

The direction is usually given by the angle θ measured from the positive x‑axis:

方向通常用相对于 x 轴正方向的夹角 θ 来表示：

θ = tan^-1(v_y / v_x)

Be careful with the quadrant when calculating θ; the arctan function on a calculator typically returns values between -90° and +90°, so you may need to add 180° or use a diagram to obtain the correct bearing.

计算 θ 时要注意象限；计算器上的反正切函数通常返回 -90° 到 +90° 之间的值，因此你可能需要加 180° 或借助示意图来确定正确的方位角。

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6. Unit Vectors | 单位向量

ngth 1 that indicates direction. In Cartesian coordinates, the standard unit vectors are î (pointing along +x), ĵ (along +y) and k̂ (along +z). Any vector can be expressed as a linear combination of these: v⃗ = v_x î + v_y ĵ + v_z k̂.

单位向量是长度为 1 且表示方向的向量。在笛卡尔坐标系中，标准单位向量为 î（指向 +x 方向）、ĵ（指向 +y 方向）和 k̂（指向 +z 方向）。任何向量都可表示为这些单位向量的线性组合：v⃗ = v_x î + v_y ĵ + v_z k̂。

To find a unit vector û in the direction of a given vector v⃗, divide the vector by its magnitude: û = v⃗ / |v⃗|.

要找到沿给定向量 v⃗ 方向的单位向量 û，将该向量除以其大小即可：û = v⃗ / |v⃗|。

Unit vectors simplify many calculations, especially when describing forces along ropes, planes or electric fields.

单位向量能简化许多计算，尤其在描述沿绳子、平面或电场方向的作用力时非常方便。

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7. Scalar (Dot) Product | 标量积（点乘）

The scalar product of two vectors a⃗ and b⃗ yields a scalar quantity. It is defined as:

两个向量 a⃗ 和 b⃗ 的标量积给出一个标量。其定义为：

a⃗ · b⃗ = |a⃗| |b⃗| cosθ

where θ is the angle between the vectors when they are placed tail‑to‑tail. In component form, for a⃗ = (a_x, a_y, a_z) and b⃗ = (b_x, b_y, b_z), the dot product is a_xb_x + a_yb_y + a_zb_z.

其中 θ 是两向量尾尾相接时的夹角。用分量形式表示时，若 a⃗ = (a_x, a_y, a_z)、b⃗ = (b_x, b_y, b_z)，点乘结果为 a_xb_x + a_yb_y + a_zb_z。

Key physical application: work done by a constant force F⃗ acting over a displacement s⃗ is W = F⃗ · s⃗ F s cosθ. When the force is perpendicular to the displacement (θ = 90°), the work is zero.

重要物理应用：恒力 F⃗ 做功位移 s⃗ 时的功为 W = F⃗ · s⃗ = F s cosθ。当力与位移垂直时（θ = 90°），做功为零。

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8. Vector (Cross) Product | 向量积（叉乘）

The cross product of two vectors a⃗ and b⃗ produces another vector that is perpendicular to both. Its magnitude is:

两个向量 a⃗ 和 b⃗ 的叉乘结果是一个垂直于该两向量的新向量。其大小为：

|a⃗ × b⃗| = |a⃗| |b⃗| sinθ

and the direction is given by the right‑hand rule: curl the fingers of your right hand from a⃗ towards b⃗, and your thumb points in the direction of a⃗ × b⃗.

方向由右手定则确定：将右手四指从 a⃗ 转向 b⃗，拇指所指即为 a⃗ × b⃗ 的方向。

In terms of components, using î, ĵ, k̂ t can be evaluated as a determinant. A vital use in physics is torque (moment of force): τ⃗ = r⃗ × F⃗, where r⃗ is the position vector from the pivot to the point of application of the force.

用分量和单位向量表示时，叉乘可通过行列式计算。物理学中一个关键应用是力矩（力的转动效果）：τ⃗ = r⃗ × F⃗，其中 r⃗ 是从支点到力作用点的位置向量。

Quantity	Expression
Torque magnitude	τ = r F sinθ
Direction	Perpendicular to r⃗ and F⃗ (right‑hand rule)

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9. Applications in Kinematics | 运动学中的应用

In kinematics, displacement, velocity and acceleration are all vectors. This means motion in two or three dimensions can be treated independently along each axis. For projectile motion, the horizontal velocity v_x remains constant (if air resistance is negligible), while the vertical motion experiences constant acceleration a_y = -g.

在运动学中，位移、速度和加速度都是向量。这意味着二维或三维运动可沿每个坐标轴独立处理。在抛体运动中，水平速度 v_x 保持不变（若忽略空气阻力），而竖直方向则经历恒定加速度 a_y = -g。

Relative velocity is another vector addition example: the velocity of A relative to B is v⃗_AB = v⃗_A - v⃗_B. Careful vector subtraction helps solve problems involving moving walkways, boats in currents, and aeroplane flight paths.

相对速度是向量加法的又一个例子：A 相对于 B 的速度为 v⃗_AB = v⃗_A - v⃗_B。仔细进行向量减法有助于解决涉及自动人行道、水流中的船只以及飞机航迹等的问题。

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10. Applications in Forces and Equilibrium | 力和平衡中的应用

ct is in translational equilibrium, the vector sum of all forces acting on it is zero: Σ F⃗ = 0. This yields two independent scalar equations for 2D problems: Σ F_x = 0 and Σ F_y = 0. Free‑body diagrams are essential for identifying all forces and resolving them into components.

当物体处于平动平衡时

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