📚 Vectors: Concepts and Representation | 矢量：概念与表示
In physics, a vector is a quantity that has both magnitude and direction. Understanding how to represent and manipulate vectors is essential for describing motion, forces, electric fields, and many other physical phenomena.
在物理学中，矢量是既有大小又有方向的量。理解如何表示和运算矢量，对于描述运动、力、电场以及许多其他物理现象至关重要。
1. Scalar vs Vector | 标量与矢量
A scalar quantity is described completely by a single number (magnitude) with appropriate units. Examples include mass, temperature, time, and energy. A vector quantity, however, requires both a magnitude and a direction. Examples include displacement, velocity, acceleration, and force.
标量只需一个带单位的数值（大小）就能完全描述，例如质量、温度、时间和能量。而矢量则必须同时给出大小和方向，例如位移、速度、加速度和力。
- Scalar: distance, speed, mass, energy | 标量：距离、速率、质量、能量
- Vector: displacement, velocity, acceleration, force | 矢量：位移、速度、加速度、力
distance = 5 m (scalar) vs displacement = 5 m north (vector)
2. Vector Notation | 矢量的符号表示
In printed materials, vectors are often shown in boldface, such as v or F. In handwriting, an arrow is placed over the symbol, such as \(\vec{v}\) — but in this text we use bold or an arrow-like Unicode where possible.
在印刷材料中，矢量常用加粗字母表示，如 v 或 F。在手写时，通常在符号上方加箭头。虽然标准手写中常写作 v⃗，但本文统一用加粗或箭头形式。
F (bold) or F⃗ (arrow) indicates a vector; F alone is a vector, while F (italic) usually denotes its magnitude.
|F⃗| = F represents the magnitude (always positive).
3. Geometric Representation | 几何表示：箭头
A vector can be drawn as an arrow. The length of the arrow represents the magnitude, and the arrowhead points in the direction of the vector. The tail is the starting point, and the head is the ending point.
矢量可以用箭头来画：箭头的长度代表大小，箭头指向代表方向。箭尾为起点，箭头为终点。
- Length ∝ magnitude | 长度正比于大小
- Orientation and arrowhead define direction | 朝向和箭头定义方向
- Equal vectors have same length and same direction | 相等矢量具有相同的长度和方向
Displacement from A to B: arrow from A (tail) to B (head)
4. Component Form | 分量表示
In a Cartesian coordinate system, a vector can be resolved into perpendicular components. A two-dimensional vector A with magnitude A and angle θ measured from the +x axis has components:
在直角坐标系中，矢量可以分解为互相垂直的分量。一个二维矢量 A，大小为 A、方向与 +x 轴的夹角为 θ，其分量为：
Aₓ = A cos θ Aᵧ = A sin θ
Conversely, given components, the magnitude and direction are found by:
反过来，已知分量时，大小和方向为：
A = √(Aₓ² + Aᵧ²) θ = tan⁻¹(Aᵧ / Aₓ)
5. Unit Vectors | 单位矢量
A unit vector has a magnitude of exactly 1 and points in a specific direction. In Cartesian coordinates, the unit vectors along the x, y, and z axes are denoted i, j, and k (or î, ĵ, k̂).
单位矢量的大小恰好为 1，用于指定方向。在直角坐标系中，沿 x、y、z 轴的单位矢量分别记为 i、j、k（或 î、ĵ、k̂）。
|i| = |j| = |k| = 1
Any vector can be written as a sum of its components multiplied by unit vectors:
任何矢量都可以写成其分量乘以单位矢量之和：
A = Aₓ i + Aᵧ j + A_z k
6. Adding Vectors Graphically | 矢量的图解加法
Two vectors can be added by placing the tail of the second vector at the head of the first. The resultant vector runs from the tail of the first to the head of the second (triangle method). Alternatively, the parallelogram method places both tails together, and the diagonal of the parallelogram gives the resultant.
两个矢量相加时，把第二个矢量的箭尾放在第一个矢量的箭头上，从第一个矢量的箭尾指向第二个矢量的箭头即为合矢量（三角形法则）。也可用平行四边形法则：让两个矢量共起点，以它们为邻边作平行四边形，对角线就是合矢量。
R = A + B
- Triangle method: tail-to-head | 三角形法则：首尾相接
- Parallelogram method: common tail, diagonal | 平行四边形法则：共起点，取对角线
- Addition is commutative: A + B = B + A | 加法满足交换律：A + B = B + A
7. Adding Vectors by Components | 矢量的分量加法
The most reliable way to add vectors is to add their corresponding components separately. If A = (Aₓ, Aᵧ) and B = (Bₓ, Bᵧ), then the resultant R = A + B has components:
最可靠的矢量加法是将各对应分量分别相加。若 A = (Aₓ, Aᵧ)，B = (Bₓ, Bᵧ)，则合矢量 R = A + B 的分量为：
Rₓ = Aₓ + Bₓ Rᵧ = Aᵧ + Bᵧ
For three-dimensional vectors, include the z-components similarly:
对于三维矢量，同样将 z 分量相加：
R_z = A_z + B_z
The magnitude of the resultant is:
合矢量的大小为：
|R| = √(Rₓ² + Rᵧ² + R_z²)
8. Subtracting Vectors | 矢量的减法
Subtracting a vector is equivalent to adding its opposite. The negative of vector A has the same magnitude but opposite direction, written as −A.
减去一个矢量等价于加上它的相反矢量。矢量 A 的负矢量 −A 大小相同，但方向相反。
A − B = A + (−B)
Subtraction by components:
分量减法：
(A − B)ₓ = Aₓ − Bₓ (A − B)ᵧ = Aᵧ − Bᵧ
Graphically, A − B can be drawn by connecting the head of B to the head of A (when tails are placed together).
作图时，若将 A 和 B 的箭尾放一起，则从 B 的箭头指向 A 的箭头即为 A − B。
9. Multiplying a Vector by a Scalar | 标量乘以矢量
When a vector A is multiplied by a scalar c, each component is multiplied by c. The magnitude becomes |c| times the original magnitude, and the direction stays the same if c is positive, or reverses if c is negative.
矢量 A 与标量 c 相乘时，每个分量都乘以 c。大小变为原来的 |c| 倍；若 c 为正，方向不变；若 c 为负，方向反转。
cA = (cAₓ) i + (cAᵧ) j
- c > 0: same direction | c > 0：方向相同
- c < 0: opposite direction | c < 0：方向相反
- c = 0: zero vector | c = 0：零矢量
10. Higher-Dimensional Vectors | 多维矢量
Vectors in IB physics are mainly two- or three-dimensional. However, the same component rules extend to any number of dimensions. In three dimensions, a vector is written as:
IB 物理中的矢量主要是二维或三维的。不过，同样的分量法则可以推广到任意维度。在三维空间中，矢量写作：
A = Aₓ i + Aᵧ j + A_z k
Its magnitude is:
其大小为：
|A| = √(Aₓ² + Aᵧ² + A_z²)
Direction in 3D is often specified by direction angles with respect to each axis, or by unit vector notation.
三维方向通常用与各坐标轴的夹角，或用单位矢量形式表示。
11. Applications in IB Physics | 在IB物理中的应用
Vectors appear in nearly every topic: resolving forces on inclined planes, adding velocity vectors in projectile motion, analysing electric field components, and combining momentum vectors in collisions.
矢量几乎出现在每个主题中：斜面上的力的分解、抛体运动中的速度矢量合成、电场分量的分析，以及碰撞中动量矢量的合成。
- Net force: summing force vectors | 合力：力的矢量求和
- Relative velocity: vector subtraction | 相对速度：矢量的差
- Projectile motion: horizontal and vertical components | 抛体运动：水平和竖直分量
- Equilibrium: resultant vector = 0 | 平衡：合矢量为零
F_net = ma (all quantities are vectors)
12. Common Mistakes and Tips | 常见错误与提示
Students often forget to include direction when stating a vector quantity, or they add magnitudes without considering directions. Always draw a sketch and resolve into components before calculating.
学生常犯的错误包括：表述矢量时忘记给出方向，或直接相加大小而不考虑方向。计算前务必画示意图并分解为分量。
- Always state magnitude and direction for a vector | 矢量必须同时说明大小和方向
- Use SOH-CAH-TOA for resolving components | 用 SOH-CAH-TOA 分解分量
- Check that units are consistent | 检查单位是否一致
- When in doubt, use component method instead of geometry | 不确定时，用分量法而非几何法
R = √(Rₓ² + Rᵧ²) and θ = tan⁻¹(Rᵧ / Rₓ)
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply