📚 Vectors and Scalars | 矢量和标量
In physics, every measurable quantity can be classified as either a scalar or a vector. Understanding the distinction between these two types is essential for mastering mechanics, kinematics, and fields. IB Physics Topic 1.3 introduces vectors and scalars, including how to combine, resolve, and manipulate them. This article provides a complete revision guide with clear explanations and worked examples.
在物理学中，每个可测量的量都可以分为标量或矢量。理解这两类量的区别是掌握力学、运动学和场的基础。IB 物理第 1.3 章介绍了矢量和标量，包括如何合成、分解和运算。本文提供一份完整的复习指南，包含清晰的解释和典型例题。
1. What are Scalars? | 什么是标量？
A scalar is a physical quantity that has magnitude (size) only. It is completely described by a number and a unit. Scalars follow ordinary arithmetic rules: if you have 5 kg of sugar and add 3 kg more, the total mass is 8 kg. Common scalar quantities include mass, time, temperature, speed, distance, energy, and electric charge.
标量是只有大小（量值）的物理量，仅用一个数值和一个单位就能完整描述。标量遵循普通算术规则：如果你有 5 kg 糖，再加 3 kg，总质量为 8 kg。常见的标量包括质量、时间、温度、速率、路程、能量和电荷。
Scalar quantities are independent of direction. For example, the distance you travel (e.g., 100 m) does not tell you where you ended up relative to your starting point. This is why we need vector quantities to describe motion fully.
标量与方向无关。例如，你行驶的路程（如 100 m）并不能说明你最终相对于起点的位置。正因如此，我们需要矢量来完整描述运动。
2. What are Vectors? | 什么是矢量？
A vector is a physical quantity that possesses both magnitude and direction. Vectors must be manipulated using vector algebra, not simple arithmetic. Examples of vector quantities include displacement, velocity, acceleration, force, momentum, and electric field strength.
矢量是同时具有大小和方向的物理量。矢量必须用矢量代数运算，而不能用简单的算术。矢量的例子包括位移、速度、加速度、力、动量和电场强度。
When we say an object moves with a velocity of 5 m/s, that is incomplete in vector terms; the direction must also be stated, for instance, 5 m/s due east. In IB Physics, vectors are often represented by boldface symbols (v) or by an arrow above the letter. In this article we use boldface for vectors.
当我们说一个物体以 5 m/s 的速度运动时，在矢量描述上是不完整的；还必须指明方向，例如 5 m/s 向正东。在 IB 物理中，矢量通常用粗体符号（v）或字母上方加箭头表示。本文中使用粗体表示矢量。
3. Representing Vectors | 矢量的表示方法
Vectors can be drawn as arrows. The length of the arrow represents the magnitude (to a chosen scale), and the arrowhead points in the direction of the vector. The starting point is called the tail and the tip is the head. For example, a vector F of 10 N acting at 30° above the horizontal can be drawn as an arrow with a length proportional to 10 N and direction at 30°.
矢量可以画成箭头。箭头的长度代表大小（按选定的比例），箭头指向矢量的方向。起点称为尾，终点称为头。例如，一个大小为 10 N、与水平方向成 30° 的力 F，可以画成与 10 N 成比例的长度且方向为 30° 的箭头。
Vectors are also expressed in component form using unit vectors. In two dimensions, any vector A can be written as A = Aₓ i + Aᵧ j, where i and j are unit vectors along the x- and y-axes, and Aₓ, Aᵧ are the scalar components.
矢量也可用分量形式结合单位矢量表示。在二维中，任意矢量 A 都可写为 A = Aₓ i + Aᵧ j，其中 i 和 j 分别是沿 x 轴和 y 轴的单位矢量，Aₓ、Aᵧ 为标量分量。
4. Addition of Vectors: Tip-to-Tail Method | 矢量加法：首尾相接法
Vectors are added geometrically by placing the tail of the second vector at the tip of the first. The resultant vector R is drawn from the tail of the first to the tip of the last vector. This is the tip-to-tail (or head-to-tail) method. For two vectors A and B, the resultant R = A + B.
矢量相加采用几何方法：将第二个矢量的尾接在第一个矢量的头上。合成矢量 R 从第一个矢量的尾画至最后一个矢量的头。这就是首尾相接法（或头尾法）。对于两个矢量 A 和 B，合矢量为 R = A + B。
If more than two vectors are added, continue placing each vector's tail at the tip of the previous one. The order does not matter; vector addition is commutative: A + B = B + A. This method is particularly useful in force diagrams and navigation problems.
如果相加两个以上的矢量，则继续将每个矢量的尾接在前一个矢量的头上。顺序并不重要；矢量加法满足交换律：A + B = B + A。该方法在受力图和导航问题中尤为实用。
5. Parallelogram Law of Vector Addition | 矢量加法的平行四边形定则
For two vectors acting at a point, the parallelogram law gives the same resultant. The two vectors are drawn tail-to-tail, forming two adjacent sides of a parallelogram. The diagonal from the common tail to the opposite corner represents the resultant R = A + B. The magnitude and direction can be found using trigonometry.
对于从同一点出发的两个矢量，平行四边形定则给出相同的合成结果。将两矢量尾尾相接，构成平行四边形的两条邻边；从公共尾到对角的对角线表示合矢量 R = A + B。其大小和方向可用三角学求解。
The magnitude of R can be calculated using the law of cosines:
R = √ A² + B² + 2AB cos θ
where θ is the angle between A and B. This formula is extremely useful when vectors are not perpendicular.
合力 R 的大小可用余弦定理计算：
R = √ A² + B² + 2AB cos θ
其中 θ 是 A 与 B 之间的夹角。当矢量不互相垂直时，该公式非常实用。
6. Subtraction of Vectors | 矢量减法
Subtracting a vector is equivalent to adding its negative. The negative of a vector has the same magnitude but opposite direction. So A − B = A + (−B). Graphically, you reverse the direction of B and then use tip-to-tail addition. Vector subtraction is essential for finding changes in velocity, relative velocity, and many other situations.
矢量减法等同于加上该矢量的负矢量。负矢量大小相同但方向相反。因此 A − B = A + (−B)。在几何表示中，你把 B 的方向反过来，然后用首尾相接法相加。矢量减法在求解速度变化量、相对速度等情境中至关重要。
7. Resolving Vectors into Components | 矢量的分解
Any vector can be broken down into perpendicular components, usually along the x- and y-axes. For a vector A of magnitude A at an angle θ to the x-axis:
Aₓ = A cos θ, Aᵧ = A sin θ
These components behave like scalars along each axis, and the vector can be reconstructed using A = Aₓ i + Aᵧ j.
任何矢量都可以分解为互相垂直的分量，通常沿 x 轴和 y 轴。对一个大小为 A、与 x 轴夹角为 θ 的矢量 A 来说：
Aₓ = A cos θ, Aᵧ = A sin θ
这些分量在各自轴上像标量一样运算，而原矢量可通过 A = Aₓ i + Aᵧ j 重新合成。
ponents to get Rₓ, and all y-components for Rᵧ. The magnitude of the resultant is then R = √Rₓ² + Rᵧ² n is φ = tan⁻¹(Rᵧ / Rₓ). This analytical method is often more accurate than scale drawing.
用分量法求多个矢量的和时，先将所有 x 分量相加得到 Rₓ，所有 y 分量相加得到 Rᵧ。合矢量的大小为 R = √Rₓ² + Rᵧ²，方向为 φ = tan⁻¹(Rᵧ / Rₓ)。这种解析方法通常比按比例画图更精确。
8. Unit Vectors | 单位矢量
Unit vectors have a magnitude of exactly 1 and point in a specific direction. In Cartesian coordinates, i, j, and k are the unit vectors along the x-, y-, and z-axes. They allow any vector to be expressed in a standard, easy-to-manipulate form: F = Fₓ i + Fᵧ j + Fz k.
单位矢量的大小恰好为 1，并指向特定方向。在直角坐标系中，i、j 和 k 分别是沿 x、y 和 z 轴的单位矢量。它们使任意矢量可用标准且易于运算的形式表达：F = Fₓ i + Fᵧ j + Fz k。
IB Physics often uses unit vectors in mechanics and fields. For example, the weight of an object can be written as W = −mg j if the y-axis is taken as upward. This compact notation simplifies vector equations.
IB 物理常在力学和场论部分使用单位矢量。例如，若取 y 轴向上，物体的重力可写为 W = −mg j。这种紧凑记法使矢量方程更加简洁。
9. Scalar (Dot) Product | 标量积（点乘）
The scalar product of two vectors A and B is defined as:
A · B = |A| |B| cos θ
where θ is the angle between them. The result is a scalar. If A and B perpendicular, cos 90° = 0 so the dot product is zero. Work done by a constant force, W = F · s, is a classic application.
两矢量 A 与 B 的标量积定义为：
A · B = |A| |B| cos θ
其中 θ 为两矢量夹角。结果为标量。如果 A 与 B 垂直，cos 90° = 0，因此点乘为零。恒力做功 W = F · s 是一个经典应用。
In component form, if A = Aₓ i + Aᵧ j and B = Bₓ i + Bᵧ j, then A · B a simple algebraic calculation used frequently when vectors are given by their components.
在分量形式下，若 A = Aₓ i + Aᵧ j 且 B = Bₓ i + Bᵧ j，则 A · B = AₓBₓ + AᵧBᵧ。当矢量以分量形式给出时，这是常用的简单代数计算。
10. Vector (Cross) Product | 矢量积（叉乘）
The vector product of two vectors A and B gives a third vector perpendicular to the plane containing A and B. Its magnitude is:
|A × B| = |A| |B| sin θ
where θ is the smaller angle between them. The direction is given by the right-hand rule: curl fingers from A to B, and the thumb points in the direction of A × B. The cross product is used for torque, angular momentum, and magnetic force.
两矢量 A 与 B 的矢量积得到第三个矢量，垂直于 A 和 B 所在平面。其大小为：
|A × B| = |A| |B| sin θ
其中 θ 为两矢量间较小夹角。方向由右手定则确定：四指从 A 弯曲指向 B，拇指所指即为 A × B 的方向。叉乘用于力矩、角动量和磁力等。
In component notation, for vectors in the xy-plane, A × B = (AₓBᵧ − AᵧBₓ) k. Note that the cross product is not commutative; in fact, A × B = −(B × A).
在分量记法中，对于 xy 平面内的矢量，A × B = (AₓBᵧ − AᵧBₓ) k。注意叉乘不满足交换律；实际上 A × B = −(B × A)。
11. Worked Example | 典型例题
A boat sails 3.0 km due east and then 4.0 km due north. Find the resultant displacement (a) graphically and (b) by component method. (c) Determine the direction relative to east.
一艘船先向正东航行 3.0 km，再向正北航行 4.0 km。求 (a) 画图法和 (b) 分量法求合位移；(c) 求相对于正东的方向。
Solution: Taking east as x-axis and north as y-axis: d₁ = 3.0 i km, d₂ = 4.0 j km. Resultant R = 3.0 i + 4.0 j km. Magnitude: R = √3.0² + 4.0² east. The tip-to-tail drawing would show a right-angled triangle confirming these values.
解：取东为 x 轴，北为 y 轴：d₁ = 3.0 i km，d₂ = 4.0 j km。合位移 R = 3.0 i + 4.0 j km。大小：R = √3.0² + 4.0² = 5.0 km。方向：θ = tan⁻¹(4.0/3.0) ≈ 53° 东偏北。首尾相接图应展示一个直角三角形，与这些数值一致。
12. Summary and IB Exam Tips | 总结与IB考试提示
Vectors and scalars are a fundamental topic that underpins nearly all of IB Physics. Always check whether a quantity has direction before applying equations. For addition and subtraction of non-collinear vectors, the component method is the most reliable technique. Memorise the cosine and sine rules for quick calculations in scalable diagrams, but be prepared to use components for precise results.
矢量和标量是贯穿几乎整个 IB 物理的基础主题。在代入方程前，务必检查一个量是否带有方向。对于非共线矢量的加减运算，分量法是最可靠的方法。记住余弦定理和正弦定理以便在比例图中快速计算，但要做好使用分量法获得精确结果的准备。
- Scalars: magnitude only; Vectors: magnitude and direction.
- 标量：只有大小；矢量：大小与方向。
- Addition: tip-to-tail or parallelogram; components often easier.
- 加法：首尾相接或平行四边形；分量法通常更简单。
- Dot product yields a scalar; cross product yields a vector perpendicular to both.
- 点乘结果为标量；叉乘结果为一个垂直于两矢量的矢量。
- In IB exams, always define your coordinate axes and be explicit about directions.
- 在 IB 考试中，务必定义坐标轴并明确注明方向。
Practice vector decomposition with trigonometric functions and the use of unit vectors i, j. Being fluent in these operations will give you confidence in kinematics, forces, and fields questions.
多练习矢量的三角函数分解以及单位矢量 i、j 的使用。熟练这些运算将让你在运动学、力和场等问题中充满信心。
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