Its main objective is to understand physical phenomena, not chemical phenomena, not biological phenomena 其主要目标是理解物理现象，而不是化学现象，也不是生物现象
physicists try to understand how 物理学家试图理解如何
nature works in physical phenomenon and how nature behaves in such ways 自然在物理现象中运作，自然以这样的方式表现
an experimental science: physicists observe physical phenomena using experimental techniques 实验科学：物理学家使用实验技术观察物理现象
Develop theories, models, laws and principles to explain the physical phenomena 发展理论、模型、定律和原则以解释物理现象
What is Physics? 物理学是什么？
"Physics" comes from the Greek word phusika, meaning "natural things"-the study of philosophy and the natural world around them “物理学”源自希腊词 phusika，意为“自然事物”——对哲学和周围自然世界的研究
Physics is a natural science based on experiments, measurements and mathematical analysis with the purpose of finding patterns that relate the phenomena of nature. 物理学是一门基于实验、测量和数学分析的自然科学，旨在寻找与自然现象相关的规律。
The patterns are called physical theories. 这些模式被称为物理理论。
A very well established theory is called a physical law or principle. 一个非常成熟的理论被称为物理定律或原则。
For example, the description of motion 例如，运动的描述
You use a ruler to measure the position and a stop watch to measure the time 你用尺子测量位置，用秒表测量时间
Plot a position and time graph to describe the motion (Kinetics) 绘制位置与时间图以描述运动（动力学）
The motion can be explained using Newton's law (Dynamics) 该运动可以用牛顿定律（动力学）来解释
AP 1201 General Physics I AP 1201 普通物理 I
Physics: a brief history 物理学：简史
Ancient Greeks studied physics as a branch of philosophy 古希腊人将物理学作为哲学的一个分支进行研究
Socrates, Plato, Aristotle and Archimedes 苏格拉底、柏拉图、亚里士多德和阿基米德
Physics became a separate field by the century 物理学在 世纪成为一个独立的领域
Galileo, Kepler and Newton helped pioneer the use of mathematics as a fundamental tool in physics 伽利略、开普勒和牛顿帮助开创了将数学作为物理学基本工具的使用
In the 1800s, the laws of electricity, magnetism and electromagnetic waves were developed: Faraday and Maxwell 在 19 世纪，电、磁和电磁波的定律被发展起来：法拉第和麦克斯韦
The laws of heat and work, and molecular origin: Kelvin, Boltzmann 热与功的定律及其分子起源：开尔文，玻尔兹曼
Modern Physics started by late to early century 现代物理学始于 世纪末到 世纪初
discovery of X-rays (Röntgen 1895), radioactivity (Becquerel 1896), the quantum hypothesis (Planck 1900), relativity (Einstein 1905) and atomic theory (Bohr 1913) X 射线的发现（伦琴 1895），放射性（贝克勒尔 1896），量子假说（普朗克 1900），相对论（爱因斯坦 1905）和原子理论（玻尔 1913）
Development of quantum mechanics revolutionized physics and allowed a better understanding of matters in the atomic, molecular, nuclear, and sub-nuclear scale 量子力学的发展彻底改变了物理学，使我们更好地理解原子、分子、核及亚核尺度的事物
Physics is the foundation of modern technologies 物理学是现代技术的基础
Engineers need to know physics 工程师需要了解物理学
Civil engineer: force, equilibrium in a bridge 土木工程师：桥梁中的力与平衡
Mechanical engineer: motion in a car engine 机械工程师：汽车发动机中的运动
Biomedical: force in artificial joint 生物医学：人工关节中的力
Electrical & electronics: current in devices, motion of a motor 电气与电子：设备中的电流，电动机的运动
Chemical engineer: entropy, reaction dynamics 化学工程师：熵，反应动力学
An Excellent Beach Book about Physics 一本关于物理的优秀海滩书籍
A CGMPLETE EDUCATIGN-WITHEUT THE TUITIGN: 完整教育——无需学费：
I T 我 T
FRIM ARISTITLE TI EINSTEIN, AND BEYIND 弗里姆·阿里斯提尔·爱因斯坦，以及超越
TロNY.RワTHMAN, PH.D. T 罗 NY.R 瓦斯曼，博士。
QVERCIME INERTIA AND LEARN HIW THIS QVERCIME 惯性并学习如何做到这一点
BRAVE NEW WIRLD WIRKS! 勇敢的新世界运作！
For all of you who break out in a sweat at the thought of thermodynamics, or freeze up at the mention of quantum mechanics, like a bolt from the blue, Instant Physics will zap you through the fascinating history of our most basic, yet baffling, science. 对于所有一想到热力学就出汗，或提到量子力学就感到紧张的人来说，《瞬间物理》将带你迅速穿越我们最基本却又令人困惑的科学的迷人历史。
From the thousand-year search for proof of the existence of the ever-elusive atom to the varied and heated arguments behind the big bang theory, Instant Physics answers all the heavy questions with a light touch. Dr. Tony Rothman explains in clear prose the inner workings of Newton's apple and Maxwell's electromagnetic waves and simultaneously offers wry observations about the state of physics in the world today. 从千年寻找永 elusive 原子的存在证明，到关于大爆炸理论的各种激烈争论，《瞬间物理》以轻松的方式回答所有重大问题。托尼·罗斯曼博士用清晰的语言解释了牛顿的苹果和麦克斯韦的电磁波的内在运作，同时对当今物理学的现状提出了尖锐的观察。
With Instant Physics you'll learn: 通过即时物理，您将学习：
How the Greek philosophers used the sledgehammer of mathematics to break apart the mysteries of the physical universe. 希腊哲学家如何利用数学的巨锤来揭开物理宇宙的奥秘。
Why gravity is a "romantic" force. 为什么引力是一种“浪漫”的力量。
Enough of Einstein's theories of relativity to discuss knowingly the derivation of . 足够的爱因斯坦相对论理论，以便能够知晓地讨论 的推导。
How to tell the difference between a gluon, a meson, and a quark, even if you can't see them. 如何区分胶子、介子和夸克，即使你看不见它们。
Instant Physics is crammed with special features, including chapter summaries, who's who lists, biographical and historical tidbits, and a host of illustrations, photos, equations, diagrams, and drawings. 《瞬间物理》充满了特别的内容，包括章节摘要、人物介绍、传记和历史小知识，以及大量插图、照片、方程式、图表和图画。
Tony Rothman, Ph.D., is an associate at Harvard College Observatory and teaches physics and astronomy at Bennington College. He is the author of A Physicist on Madison Avenue and Science à la Mode. 托尼·罗斯曼，博士，是哈佛大学天文台的副研究员，并在本宁顿学院教授物理和天文学。他是《麦迪逊大道上的物理学家》和《时尚科学》的作者。
INSTANT l HVEICE 即时 l HVEICE
PRICE 价格
ISBN 0-449-90693-3
NSTANT PHYSICS ||II|||||||||||||||||| |||||||| 51195 瞬时物理学 ||II|||||||||||||||||| |||||||| 51195
Physics as it was developing by a Nobel Laureate 诺贝尔奖得主所发展的物理学
AP 1201 General Physics I AP 1201 普通物理 I
Topics for Chapter 1 第一章主题
Units, physical quantities 单位，物理量
Dimensional analysis 维度分析
Vectors and Scalars 向量和标量
add vectors graphically 图形化地添加向量
vector components 向量分量
unit vectors and components 单位向量和分量
multiplying vectors 向量相乘
A little calculus: 一点微积分：
some basic derivatives (differentiation) of functions 一些函数的基本导数（微分）
Basic concept of integration 积分的基本概念
Standards and Units* 标准和单位*
Length, time, and mass are three fundamental quantities of physics. 长度、时间和质量是物理学的三个基本量。
The International System (SI for Système International) is the most widely used system of units. 国际单位制（SI，法语为 Système International）是使用最广泛的单位制。
In SI units, length is measured in meters, time in seconds, and mass in kilograms. 在国际单位制中，长度以米为单位，时间以秒为单位，质量以千克为单位。
Units of measurement* 计量单位*
Dimension 维度
SI (mks) Unit 国际单位制（米千克秒）单位
Definition 定义
TABLE 1.4 表 1.4
Length 长度
meters (m) 米 (m)
光在 秒内行驶的距离
Distance traveled by
light in
second
一些用于“公制”（SI 和 cgs）单位的 10 的幂的前缀
Some Prefixes for Powers of
10 Used with "Metric" (SI
and cgs) Units
Power 权力
Prefix 前缀
Abbreviation 缩写
atto- 阿托-
a
Mass 弥撒
kilogram (kg) 千克 (kg)
Mass of a specific 特定的质量
pico- 皮克
kilogram (kg) 千克 (kg)
platinum-iridium alloy 铂铱合金
纳米- 微米-
nano-
micro-
cylinder kept by Intl. 国际保留的气缸
milli- 毫-
m
Bureau of Weights and 计量局
centi- 厘-
c
Measures at Sèvres, 塞夫尔的措施，
deci- 分之一
d
France 法国
deka- 德卡-
Time 时间
seconds (s) 秒 (s)
铯-133 原子进行 9,192,631,700 次完整振荡所需的时间
Time needed for a
cesium-133 atom to
perform 9,192,631,700
complete oscillations
mega- 兆-
M
giga- 千兆-
G
tera- 特拉-
T
皮塔- 艾克萨-
peta-
exa-
P
exa-
Cole E 科尔·E
Dimensional Analysis* 维度分析*
An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you're adding "apples to apples.") 一个方程必须在维度上保持一致。要相加或相等的项必须始终具有相同的单位。（确保你是在“加苹果对苹果。”）
Always carry units through calculations. 始终在计算中保留单位。
Convert to standard units as necessary. 根据需要转换为标准单位。
Dimensions & units can be treated algebraically. 维度和单位可以用代数方式处理。
Variable from Eq. 从方程中得出的变量。
x
m
t
dimension 维度
L
M
T
Dimensional Analysis* 维度分析*
Checking equations with dimensional analysis: 通过维度分析检查方程：
Each term must have same dimension 每个项必须具有相同的维度
Two variables cannot be added if dimensions are different 两个变量的维度不同则无法相加
Multiplying variables is always fine 乘法变量总是可以的
Numbers (e.g. 1/2 or ) are dimensionless 数字（例如 1/2 或 ）是无量纲的
Example 1.1* 示例 1.1*
Check the equation for dimensional consistency: 检查方程的维度一致性：
Here, is a mass, g is an acceleration, v and are velocities, is a length 这里， 是一个质量，g 是加速度，v 和 是速度， 是长度
Example 1.2* 示例 1.2*
Consider the equation: 考虑方程：
Where and are masses, is a radius and is a velocity. What are the dimensions of ? 其中 和 是质量， 是半径， 是速度。 的维度是什么？
Units vs. Dimensions* 单位与维度*
Dimensions: L, T, M, L/T ... 尺寸：L，T，M，L/T ...
Units: m, mm, cm, kg, g, mg, s, hr, years ... 单位：米，毫米，厘米，千克，克，毫克，秒，小时，年……
When an equation is all algebraic: check dimensions 当一个方程是全代数时：检查维度
When numbers are inserted: check units 当插入数字时：检查单位
Units obey same rules as dimensions: Never add terms with different units 单位遵循与维度相同的规则：绝不要将不同单位的项相加
Angles are dimensionless but have different units (degrees or radians) 角度是无量纲的，但有不同的单位（度或弧度）
In physics or never occur unless Y is dimensionless 在物理学中， 或 只有在 Y 是无量纲时才会发生
Two kinds of quantities 两种数量
In physics, we must handle quantities, such as your weight, your height. 在物理学中，我们必须处理数量，例如你的体重和身高。
Quantities can be obtained from experiments. 数量可以通过实验获得。
Quantities are usually represented by numbers 数量通常用数字表示
There are two kinds of quantities: scalar and vector 有两种量：标量和向量
Vectors and scalars 向量和标量
A scalar is a quantity represented by a single number, such as your weight or height 标量是由一个单一数字表示的量，例如你的体重或身高
A vector is a quantity having both a magnitude and a direction. 向量是具有大小和方向的量。
a vector is usually represented by a letter with an arrow over it: or or simply . 一个向量通常用一个字母上面加一个箭头来表示： 或 或简单地 。
Magnitude of is represented by or . 的大小由 或 表示。
A typical example of vector: position of Kowloon Tong station relative to the Central station. 一个典型的向量例子：九龙塘站相对于中环站的位置。
Vector (relative position) 向量（相对位置）
The relative position of Kowloon Tong with respect to Central is a vector. 九龙塘相对于中环的位置是一个向量。
It tells you how to get to Kowloon Tong from central: 它告诉你如何从市中心到达九龙塘：
Walk a distance along a certain direction 沿着某个方向走一段距离
If the direction is changed, you go to the wrong place (red arrow) if you go for the same 如果方向改变，你会去错误的地方（红箭头），如果你走的是相同的方向
distance 距离
Other vector physical quantities 其他矢量物理量
Apart from displacement, the following physical quantities are also vectors: 除了位移，以下物理量也是向量：
Velocity, momentum (come from position) 速度，动量（来自位置）
Angular velocity, angular momentum (come from position) 角速度，角动量（来自位置）
Acceleration, force 加速度，力
electric field, magnetic field 电场，磁场
They are usually related to motion and force 它们通常与运动和力有关
Vector representation 向量表示
Graphically a vector is represented by a line with an arrowhead at its tip. 图形上，向量用一条带箭头的线表示。
The length of the line represents the vector's magnitude. 线的长度表示向量的大小。
The arrow direction represents the vector's direction. 箭头方向表示向量的方向。
The red vector is different from the black one although their magnitudes are the same 红色向量与黑色向量不同，尽管它们的大小相同
A vector can be written in terms of the coordinates: 一个向量 可以用 坐标表示：
Magnitude can be represented by a number in the figure 大小可以用图中的数字 表示
Direction can represented by the angle made with a fixed direction, angle made with the x-axis, so that 方向可以通过与固定方向形成的角度来表示，角度 与 x 轴形成，因此
Adding two vectors graphically 图形化地添加两个向量
Two or more vectors may be added graphically 两个或多个向量可以通过图形方式相加
Vector addition is commutative, i.e. does not depend on the order 向量加法是交换的，即不依赖于顺序。
(a)
(b)
(c)
Vector Addition: head to tail method 向量加法：头尾法
Adding 2 vectors by placing them head to tail, the resultant vector can be drawn from the origin to the end 将两个向量首尾相接相加，结果向量可以从原点绘制到末端
It is independent on the order, adding in different order gives the same result 它与顺序无关，以不同的顺序添加会得到相同的结果
Same result can be achieved by constructing a parallelogram with the vector pointing to the vertex between vectors and 通过构造一个平行四边形，使用指向向量 和 之间顶点的向量，可以实现相同的结果
Several vectors can be added using the head to tail method 可以使用头尾法将多个向量相加
Vector Subtraction 向量减法
Vector subtraction is similar to vector addition 向量减法类似于向量加法
Multiplying a vector by a scalar 将向量乘以标量
If is a scalar, the product has magnitude . 如果 是一个标量，则乘积 的大小为 。
Multiplying a vector by a positive scalar changes the magnitude of the 将一个向量乘以一个正标量会改变其大小
vector but not he direction 向量但不是方向
Multiplying a vector by a negative scalar reverses the direction and changes the magnitude of the vector 将一个向量乘以一个负标量会反转向量的方向并改变其大小
Components of a vector 向量的组成部分
Adding vectors graphically may not be accurate and is not convenient 图形化地相加向量可能不准确且不方便
Vector components provide a general method for adding vectors. 向量分量提供了一种添加向量的通用方法。
Any vector can be represented by an -component and a component . and directions are usually perpendicular. 任何向量都可以由一个 -分量 和一个 分量 表示。 和 方向通常是垂直的。
Use trigonometry to find the components of a vector: and , where is measured from the -axis toward the -axis. 使用三角学找到一个向量的分量： 和 ，其中 是从 轴测量到 轴的。
Magnitude of the vector : 向量 的大小：
Positive and negative components 正负组件
The components of a vector can be positive or negative numbers, as shown in the figure. 向量的分量可以是正数或负数，如图所示。
If the component is pointing to the negative direction, then the component is a negative number 如果分量指向负方向，那么该分量是一个负数
For both component vectors point in the +ve directions 对于 ，两个分量向量都指向正方向
pointing in the direction 指向 方向 pointing in the +y direction pointing in the direction 指向 +y 方向 指向 方向 pointing in the -y direction 指向 -y 方向
Example 示例
Given: 给定：
Given: 给定：
A vector in 3-D 三维中的向量
Similar to the 2-D case, in 3-D a vector can be represented by an -component , a -component and a component 类似于二维情况，在三维中，一个向量 可以由一个 -维 、一个 -维 和一个 维 表示
Unit vectors: units for Cartesian coordinates 单位向量：笛卡尔坐标的单位
A vector can be expressed by its components and in 3 dimensions along the and z axes 一个向量 可以通过其在 和 方向上的分量在三维空间中表示，沿着 和 z 轴
Unit vectors are dimensionless vectors along the axes with length=1, i.e. just indicating the direction 单位向量是沿着 轴的无量纲向量，长度为 1，即仅表示方向。
For two dimension, we need two unit vectors. For three dimension, we need three unit vectors. 对于二维，我们需要两个单位向量。对于三维，我们需要三个单位向量。
The unit vector points in the direction, points in the -direction, and points in the -direction 单位向量 指向 方向， 指向 方向， 指向 方向
All vectors can be expressed in terms of unit vectors 所有向量都可以用单位向量表示
Vector Algebra using Components 使用分量的向量代数
Vectors can be added easily using their components 向量可以通过它们的分量轻松相加
We can use the components of a set of vectors to find the components of their sum: 我们可以使用一组向量的分量来找到它们的和的分量：
The components of the sum vector is the sum of the components of all the vectors.. 和向量的分量是所有向量分量的总和。
Using unit vector representation: 使用单位向量表示：
Laws of Vector Algebra* 向量代数法则*
If are vectors and are scalars 如果 是向量， 是标量
Commutative law of Addition: 加法的交换律：
Associative Law of Addition: 加法的结合律：
Commutative Law for Multiplication: 乘法的交换律：
Associative Law for Multiplication: 乘法的结合律：
Distributive Law: 分配律：
These laws do not apply to multiplication of vectors (only a vector and scalars) 这些定律不适用于向量的乘法（仅适用于向量和标量）
There are two types of vector multiplications: scalar product (or dot product) and vector product (or the cross product) 有两种类型的向量乘法：标量积（或点积）和向量积（或叉积）
The Scalar Product 标量积
The scalar product of 2 vectors: read as A dot B (also called the "dot product") and is defined as 两个向量的标量积： 读作 A 点 B（也称为“点积”），定义为
where is the angle between them 其中 是它们之间的角度
The product is a scalar so it is called scalar product 该产品是一个标量，因此称为标量积
It is useful in calculating work done of a force 它在计算力所做的功时很有用
A scalar product can be negative depending on the subtended angle 标量积可以是负的，这取决于夹角
The scalar product is zero if the 2 vectors are perpendicular 如果两个向量垂直，则标量积为零
Calculating a scalar product 计算标量积
The following laws are valid: 以下法律有效：
commutative law for dot product 点积的交换律
In terms of components: 在组件方面：
Later you can see work done is the dot product (scalar product) of force and displacement 后来你可以看到所做的工作是力和位移的点积（标量积）
The Vector Product 向量积
The vector product of and is a vector "cross product" 和 的向量积是向量 "叉积"
The magnitude of a cross product 叉积的大小
The direction of a cross product is the plane of and such that and form a right-hand system 叉积的方向是 平面 和 ，使得 和 形成一个右手坐标系
Vector Product* 向量积*
For vector product, the following laws are valid: 对于向量积，以下定律是有效的： commutative law fails (anticommutative) 交换律失效（反交换） distributive law 分配律 associative Law 结合律
Later we will see that angular momentum is the vector product of momentum and a position vector 稍后我们将看到角动量是动量和位置向量的矢量积
Torque is the vector product of force and a position vector 扭矩是力与位置向量的矢量积
A Little Calculus 微积分入门
In mechanics we need to know some calculus for calculating velocity and acceleration 在力学中，我们需要了解一些微积分，以计算速度和加速度
We need to know how to do differentiation and integration 我们需要知道如何进行微分和积分
In this course, we use a little bit of calculus to understand motion, there are some questions in computer exercises. This is a required skill. 在本课程中，我们使用一点微积分来理解运动，计算练习中有一些问题。这是一项必备技能。
A little Calculus: Differentiation 微积分基础：微分
In mechanics some calculus is needed for calculating velocity and acceleration 在力学中，需要一些微积分来计算速度和加速度
Calculus is the mathematical study of change 微积分是对变化的数学研究
Differentiation concerns the study of the instantaneous rate of change of a variable with respect to another variable, such as growth rate 微分涉及研究一个变量相对于另一个变量的瞬时变化率，例如增长率
For example, changes with , the rate of change of with respect to represents how fast changes with . 例如， 随着 变化， 相对于 的变化率表示 随着 变化的速度。
Instantaneous means the rate of change at a particular point 瞬时意味着在特定点的变化率
Take a function 取一个函数
The change of with respect to at 2 points and is 相对于 在两个点 和 的变化是
Since this is a straight line, the rate of change at any 2 points are the same, i.e. 由于这是直线，任何两个点的变化率都是相同的，即。
The rate of change of a linear function is a constant 线性函数的变化率是一个常数
Differentiation 差异化
Take a more complicated function 采取一个更复杂的函数
Average rate of change between points and is 在点 和 之间的平均变化率是
As we decrease this average rate also changes until approaches 0 then we can have the instantaneous rate of change at point 随着我们减少 ，这个平均速率也会变化，直到 接近 0，然后我们可以在点 获得瞬时变化率
Mathematically, 在数学上，
Fig. 1 图 1
Differentiation 差异化
Consider a function 考虑一个函数
The differentiation (derivative) of with respect to is defined as 对 的导数定义为
If is time and is distance, then the derivative is the rate of change of distance with 如果 是时间， 是距离，那么导数 是距离随时间变化的速率
Fig. 1 time or speed (instantaneous) 图 1 时间或速度（瞬时）
The derivative of a function can be written as 函数 的导数可以写成
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant 一个多变量函数的偏导数是其对其中一个变量的导数，而其他变量保持不变
Take a function with two variables: so that 取一个有两个变量的函数： 使得
We can keep one variable or constant and differentiate with respect to the other variable or . 我们可以保持一个变量 或 不变，并对另一个变量 或 进行求导。
Integration 整合
Integration is the reverse of differentiation: 积分是微分的反过程 is the differentiation (derivative) of , 是 的导数
then is the integration of with respect to 然后 是关于 的 的积分
is called the indefinite integral of w.r.t 被称为相对于 的 的不定积分
But: 但是：
So and , 所以 和 ，
Example: 示例：
Definite integral 定积分
Definite integral of with respect to between and is 关于 在 和 之间的 的定积分是
area under the curve f between and 曲线 f 在 和 之间的面积
Integration of a curve from point a to point b represents the area under the curve 从点 a 到点 b 的曲线积分表示曲线下方的面积
Definite integral 定积分
Cut the area under the curve into small slices, the total area is 将曲线下方的区域切成小片，总面积为
Integration 整合
What you expect to know and understand for integration in this course: 您在本课程中期望了解和掌握的整合内容：
The meaning of integration: 整合的意义：
Reverse of differentiation 微分的逆运算
Area under a curve 曲线下的面积
Examples
from University Physics (Young and Freedman, edition Pearson) 示例
来自《大学物理》（杨和弗里德曼，第 版，皮尔森）
We may not have time to go through all the examples 我们可能没有时间逐一查看所有示例
Addition of two vectors at right angles 两个垂直向量的相加
First add the vectors graphically. 首先图形化地添加向量。
Then use trigonometry to find the magnitude and direction of the sum. 然后使用三角学来找出和的大小和方向。
Follow Example 1.5.) 遵循示例 1.5。
Example 1.6 Addition of two vectors at right angles 示例 1.6 直角相加的两个向量
A cross-country skier skis 1.00 km north and then 2.00 km east on a horizontal snowfield. How far and in what direction is she from the starting point? 一名越野滑雪者在水平的雪地上向北滑行 1.00 公里，然后向东滑行 2.00 公里。她离起点有多远，方向是什么？
SOLUTION 解决方案
IDENTIFY and SET UP: The problem involves combining two displacements at right angles to each other. In this case, vector addition amounts to solving a right triangle, which we can do using the Pythagorean theorem and simple trigonometry. The target variables are the skier's straight-line distance and direction from her starting point. Figure 1.16 is a scale diagram of the two displacements and the resultant net displacement. We denote the direction from the starting point by the angle (the Greek letter phi). The displacement appears to be about 2.4 km . Measurement with a protractor indicates that is about . 识别和设置：这个问题涉及将两个相互垂直的位移结合在一起。在这种情况下，向量加法相当于解决一个直角三角形，我们可以使用毕达哥拉斯定理和简单的三角函数来完成。目标变量是滑雪者从起点出发的直线距离和方向。图 1.16 是两个位移及其结果净位移的比例图。我们用角度 （希腊字母 phi）表示从起点的方向。位移似乎约为 2.4 公里。用量角器测量表明 约为 。
EKECUTE: The distance from the starting point to the ending point is equal to the length of the hypotenuse: EKECUTE：起点到终点的距离等于斜边的长度：
A little trigonometry (from Appendix B) allows us to find angle : 一些三角学（来自附录 B）使我们能够找到角度 ：
We can describe the direction as east of north or north of east. 我们可以将方向描述为 北偏东或 东偏北。
EVALUATE: Our answers ( 2.24 km and ) are close to our predictions. In the more general case in which you have to add two vectors not at right angles to each other, you can use the law of cosines in place of the Pythagorean theorem and use the law of sines to find an angle corresponding to in this example. (You'll find these trigonometric rules in Appendix B.) We'll see more techniques for vector addition in Section 1.8. 评估：我们的答案（2.24 公里和 ）接近我们的预测。在更一般的情况下，当你需要相加两个不成直角的向量时，可以使用余弦定理代替勾股定理，并使用正弦定理找到与本例中的 对应的角度。（你可以在附录 B 中找到这些三角函数规则。）我们将在第 1.8 节看到更多的向量加法技巧。
Example 1. Finding components 示例 1. 查找组件
(a) What are the - and -components of vector in Fig. 1.19a? The magnitude of the vector is , and the angle . (b) What are the - and -components of vector in Fig. 1.19b? The magnitude of the vector is , and the angle . (a) 图 1.19a 中向量 的 -分量和 -分量是什么？向量的大小是 ，角度是 。 (b) 图 1.19b 中向量 的 -分量和 -分量是什么？向量的大小是 ，角度是 。
SOLUTION 解决方案
IDENTIFY and SET UP: We can use Eqs. (1.6) to find the components of these vectors, but we have to be careful: Neither of the angles or in Fig. 1.19 is measured from the -axis toward the -axis. We estimate from the figure that the lengths of the com- 识别和设置：我们可以使用方程（1.6）来找到这些向量的分量，但我们必须小心：图 1.19 中的角 或 都不是从 轴测量到 轴的。我们从图中估计出 com-的长度。
ponents in part (a) are both roughly 2 m , and that those in part (b) are 3 m and 4 m . We've indicated the signs of the components in the figure. 部分(a)中的分量大约都是 2 米，而部分(b)中的分量分别是 3 米和 4 米。我们在图中标出了分量的符号。
EMECUTE: (a) The angle (the Greek letter alpha) between the positive -axis and is measured toward the negative -axis. The angle we must use in Eqs. (1.6) is . We then find EMECUTE: (a) 正 轴与 之间的角度 （希腊字母α）是朝向负 轴测量的。我们在方程（1.6）中必须使用的角度是 。然后我们发现
Had you been careless and substituted for in Eqs. (1.6), your result for would have had the wrong sign. 如果你不小心在方程 (1.6) 中将 替换为 ，那么你得到的 的结果将会是错误的符号。
(b) The - and -axes in Fig. 1.19b are at right angles, so it doesn't matter that they aren't horizontal and vertical, respectively. But to use Eqs. (1.6), we must use the angle . Then we find (b) 图 1.19b 中的 轴和 轴是垂直的，因此它们不是水平和垂直的也无所谓。但要使用方程(1.6)，我们必须使用角度 。然后我们发现
EVALUATE: Our answers to both parts are close to our predictions. But ask yourself this: Why do the answers in part (a) correctly have only two significant figures? 评估：我们对两个部分的答案都接近我们的预测。但问问自己：为什么部分（a）的答案正确地只有两个有效数字？
IDENTIFY the relevant concepts: Decide what the target variable is. It may be the magnitude of the vector sum, the direction, or both. 确定相关概念：决定目标变量是什么。它可能是向量和的大小、方向或两者。
fill SET UP the problem: Sketch the vectors being added, along with suitable coordinate axes. Place the tail of the first vector at the origin of the coordinates, place the tail of the second vector at the head of the first vector, and so on. Draw the vector sum from the tail of the first vector (at the origin) to the head of the last vector. Use your sketch to estimate the magnitude and direction of . Select the mathematical tools you'll use for the full calculation: Eqs. (1.6) to obtain the components of the vectors given, if necessary, Eqs. (1.11) to obtain the components of the vector sum, Eq. (1.12) to obtain its magnitude, and Eqs. (1.8) to obtain its direction. 填充设置问题：绘制被相加的向量，以及合适的坐标轴。将第一个向量的尾部放在坐标原点，将第二个向量的尾部放在第一个向量的头部，依此类推。从第一个向量的尾部（在原点）到最后一个向量的头部绘制向量和 。利用你的草图估计 的大小和方向。选择你将用于完整计算的数学工具：方程（1.6）以获取给定向量的分量（如有必要），方程（1.11）以获取向量和的分量，方程（1.12）以获取其大小，以及方程（1.8）以获取其方向。
EXECUTE the solution as follows: 执行解决方案如下：
Find the - and -components of each individual vector and record your results in a table, as in Example 1.7 below. If a vector is described by a magnitude and an angle , measured 找到每个单独向量的 和 分量，并将结果记录在表格中，如下面的示例 1.7 所示。如果一个向量由大小 和角度 描述，测量
from the -axis toward the -axis, then its components are given by Eqs. 1.6: 从 轴朝向 轴，则其分量由方程 1.6 给出：
If the angles of the vectors are given in some other way, perhaps using a different reference direction, convert them to angles measured from the -axis as in Example 1.6 above. 如果向量的角度以其他方式给出，可能使用不同的参考方向，请将它们转换为从 轴测量的角度，如上面的示例 1.6 所示。
2. Add the individual -components algebraically (including signs) to find , the -component of the vector sum. Do the same for the -components to find . See Example 1.7 below. 2. 将各个 -分量代数相加（包括符号）以找到 ，这是向量和的 -分量。对 -分量做同样的操作以找到 。请参见下面的例子 1.7。
3. Calculate the magnitude and direction of the vector sum using Eqs. (1.7) and (1.8): 3. 使用公式 (1.7) 和 (1.8) 计算向量和的大小 和方向 ：
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EVALUATE your answer: Confirm that your results for the magnitude and direction of the vector sum agree with the estimates you made from your sketch. The value of that you find with a calculator may be off by ; your drawing will indicate the correct value. 评估你的答案：确认你对向量和的大小和方向的结果与从草图中得出的估计一致。你用计算器找到的 的值可能偏差 ；你的图示将指示正确的值。
Adding vectors using their components-Figure 1.22 使用分量相加向量-图 1.22
Three players on a reality TV show are brought to the center of a large, flat field. Each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each contestant) the following three displacements: 三名现实电视节目中的选手被带到一个大而平坦的场地中央。每人都被给予一根米尺、一只指南针、一台计算器、一把铲子，以及（每位参赛者的顺序不同）以下三个位移：
The three displacements lead to the point in the field where the keys to a new Porsche are buried. Two players start measuring immediately, but the winner first calculates where to go. What does she calculatiti?? 三个位移指向场地中埋藏新保时捷钥匙的地点。两名玩家立即开始测量，但获胜者首先计算该去哪里。她计算了什么？
SOLUTION 解决方案
IDENTIFY and SET UP: The goal is to find the sum (resultant) of the three displacements, so this is a problem in vector addition. Figure 1.22 shows the situation. We have chosen the -axis as east and the -axis as north. We estimate from the diagram that the vector sum is about west of north (which corresponds to ). 识别和设置：目标是找到三个位移的和（结果），因此这是一个向量加法的问题。图 1.22 显示了情况。我们选择 轴为东， 轴为北。我们从图中估计向量和 大约在北偏 西（这对应于 ）。
EXECUTE: The angles of the vectors, measured from the -axis toward the -axis, are , and , respectively. We may now use Eqs. (1.6) to find the components of : 执行：从 轴测量到 轴的向量角度分别为 、 和 。我们现在可以使用公式(1.6)来找到 的分量：
We've kept an extra significant figure in the components; we'll round to the correct number of significant figures at the end of our calculation. The table below shows the components of all the displacements, the addition of the components, and the other calculations. 我们在各个分量中保留了一个额外的有效数字；我们将在计算结束时四舍五入到正确的有效数字。下表显示了所有位移的分量、分量的加法以及其他计算。
Distance 距离
[
]
Comparing to Fig. 1.22 shows that the calculated angle is clearly off by . The correct value is , or west of north. 与图 1.22 相比，计算出的角度明显偏差了 。正确值是 ，或 西偏北。
EVALUATE: Our calculated answers for and agree with our estimates. Notice how drawing the diagram in Fig. 1.22 made it easy to avoid a error in the direction of the vector sum. 评估：我们对 和 的计算结果与我们的估计一致。注意在图 1.22 中绘制图形如何使我们容易避免在向量和的方向上出现 错误。
Example 1.8 A simple vector addition in three dimensions 例 1.8 三维中的简单向量加法
After an airplane takes off, it travels 10.4 km west, 8.7 km north, and 2.1 km up. How far is it from the takeoff point? 飞机起飞后，向西飞行 10.4 公里，向北飞行 8.7 公里，向上飞行 2.1 公里。它离起飞点有多远？
SOLUTION 解决方案
Let the -axis be east, the -axis north, and the -axis up. Then the components of the airplane's displacement are , and . From Eq. (1.12), the magnitude of the displacement is 让 轴为东， 轴为北， 轴为上。则飞机位移的分量为 和 。根据公式(1.12)，位移的大小为
Example 1.8 Using unit vectors 示例 1.8 使用单位向量
Given the two displacements 给定两个位移
find the magnitude of the displacement . 找到位移的大小 。
SOLUTION 解决方案
IDENTIFY and SET UP: We are to multiply the vector by 2 (a scalar) and subtract the vector from the result, so as to obtain the vector . Equation (1.9) says that to multiply by 2 , we multiply each of its components by 2 . We can use Eq. (1.17) to do the subtraction; recall from Section 1.7 that subtracting a vector is the same as adding the negative of that vector. 识别和设置：我们要将向量 乘以 2（一个标量），并从结果中减去向量 ，以获得向量 。方程 (1.9) 表示要将 乘以 2，我们需要将其每个分量都乘以 2。我们可以使用方程 (1.17) 来进行减法；回想一下第 1.7 节，减去一个向量等同于加上该向量的负值。
EXECUTE: We have 执行：我们有
From Eq. (1.12) the magnitude of is 根据公式 (1.12)， 的大小为
EVALUATE: Our answer is of the same order of magnitude as the larger components that appear in the sum. We wouldn't expect our answer to be much larger than this, but it could be much smaller. 评估：我们的答案与出现在总和中的较大组件的数量级相同。我们不期望我们的答案会比这个大得多，但它可能会小得多。
Calculating a scalar product 计算标量积
Example 1.10 Calculating a scalar product 示例 1.10 计算标量积
Find the scalar product of the two vectors in Fig. 1.27. The magnitudes of the vectors are and . 找到图 1.27 中两个向量的标量积 。这两个向量的大小分别为 和 。
SOLUTION 解决方案
IDENTIFY and SET UP: We can calculate the scalar product in two ways: using the magnitudes of the vectors and the angle between them (Eq. 1.18), and using the components of the vectors (Eq. 1.21). We'll do it both ways, and the results will check each other. 识别和设置：我们可以通过两种方式计算标量积：使用向量的大小和它们之间的角度（公式 1.18），以及使用向量的分量（公式 1.21）。我们将两种方法都进行计算，结果将相互验证。
EXECUTE: The angle between the two vectors is , so Eq. (1.18) gives us 执行：两个向量之间的角度是 ，因此公式 (1.18) 给我们提供了
To use Eq. (1.21), we must first find the components of the vectors. The angles of and are given with respect to the -axis and are measured in the sense from the -axis to the -axis, so we can use Eqs. (1.6): 要使用公式(1.21)，我们必须首先找到向量的分量。 和 的角度是相对于 轴给出的，并且是从 轴到 轴的方向测量的，因此我们可以使用公式(1.6)：
Finding the angle between 2 vectors using the scalar product 使用标量积找到两个向量之间的角度
Example 1.11 Finding an angle with the scalar product . 例 1.11 使用标量积找到一个角度。
Find the angle between the vectors 找到向量之间的角度
SOLUTION 解决方案
IDENTIFY and SET UP: We're given the -, -, and -components of two vectors. Our target variable is the angle between them (Fig. 1.28). To find this, we'll solve Eq. (1.18), , for in terms of the scalar product and the magnitudes and . We can evaluate the scalar product using Eq. (1.21), 识别和设置：我们得到了两个向量的 、 和 分量。我们的目标变量是它们之间的角度 （图 1.28）。为了找到这个角度，我们将根据标量积 和大小 和 ，求解方程(1.18)， 中的 。我们可以使用方程(1.21)来评估标量积。 , and we can find and using Eq. (1.7). ，我们可以使用公式 (1.7) 找到 和 。
EXECUTE: We solve Eq. (1.18) for and write using Eq. (1.21). Our result is 执行：我们对方程 (1.18) 进行求解，得到 ，并使用方程 (1.21) 写出 。我们的结果是
We can use this formula to find the angle between any two vectors and . Here we have , and , and , and . Thus 我们可以使用这个公式来找到任意两个向量 和 之间的角度。这里我们有 ，和 ，和 ，和 。因此
EVALUATE: As a check on this result, note that the scalar product is negative. This means that is between and (see Fig. 1.26), which agrees with our answer. 评估：作为对该结果的检查，请注意标量积 是负的。这意味着 在 和 之间（见图 1.26），这与我们的答案一致。
Calculating the vector product-Figure 1.32 计算向量积 - 图 1.32
Use to find the magnitude and the right-hand rule to find the direction. 使用 来找到大小，并使用右手法则来确定方向。
Example 1.12 Calculating a vector product 示例 1.12 计算向量积
Vector has magnitude 6 units and is in the direction of the -axis. Vector has magnitude 4 units and lies in the -plane, making an angle of with the -axis (Fig. 1.32). Find the vector product . 向量 的大小为 6 个单位，方向沿着 轴。向量 的大小为 4 个单位，位于 平面内，与 轴成 角（图 1.32）。求向量积 。
SOLUTION 解决方案
IDENTIFY and SET UP: We'll find the vector product in two ways, which will provide a check of our calculations. First we'll use Eq. (1.22) and the right-hand rule; then we'll use Eqs. (1.27) to find the vector product using components. 识别和设置：我们将通过两种方式找到向量积，这将对我们的计算进行检查。首先，我们将使用公式（1.22）和右手法则；然后，我们将使用公式（1.27）通过分量找到向量积。
EXECUTE: From Eq. (1.22) the magnitude of the vector product is 执行：从公式 (1.22) 中，向量积的大小为
By the right-hand rule, the direction of is along the -axis (the direction of the unit vector ), so we have . 根据右手法则， 的方向沿着 轴（单位向量 的方向），因此我们有 。
To use Eqs. (1.27), we first determine the components of and : 为了使用方程 (1.27)，我们首先确定 和 的分量：
Then Eqs. (1.27) yield 然后方程 (1.27) 得到
Thus again we have . 因此我们再次得到了 。
EVALUATE: Both methods give the same result. Depending on the situation, one or the other of the two approaches may be the more convenient one to use. 评估：这两种方法得出的结果相同。根据情况，可能其中一种方法更方便使用。
