Principles of Mechanics pp 115  Cite as
Units and Vectors
 14k Downloads
Abstract
Physics is an exciting adventure that is concerned with unraveling the secrets of nature based on observations and measurements and also on intuition and imagination. Its beauty lies in having few fundamental principles being able to reach out to incorporate many phenomena from the atomic to the cosmic scale. It is a science that depends heavily on mathematics to prove and express theories and laws and is considered to be the most fundamental of physical sciences. Astronomy, geology, and chemistry all involve applications of physics’ principles and concepts. Physics doesn’t only provide theories, but it also provides techniques that are used in every area of life. Modern physical techniques were the major contributors to the wealth of mankind’s knowledge in the past century.
1.1 Introduction
Physics is an exciting adventure that is concerned with unraveling the secrets of nature based on observations and measurements and also on intuition and imagination. Its beauty lies in having few fundamental principles being able to reach out to incorporate many phenomena from the atomic to the cosmic scale. It is a science that depends heavily on mathematics to prove and express theories and laws and is considered to be the most fundamental of physical sciences. Astronomy, geology, and chemistry all involve applications of physics’ principles and concepts. Physics doesn’t only provide theories, but it also provides techniques that are used in every area of life. Modern physical techniques were the major contributors to the wealth of mankind’s knowledge in the past century.
A simple law in physics can be used to explain a wide range of complex phenomena that may appear to be not related. When studying a complex physical system, a simplified model of the system is usually used, where the minor effects are neglected and the main features of the system are concentrated upon. For example, when dealing with an object falling near the earth’s surface, air resistance can be neglected. In addition, the earth is usually assumed to be spherical and homogeneous. However, in reality, the earth is an ellipsoid and is not homogeneous. The difference between the calculations of these different models can be assumed to be insignificant.
Physics can be divided into two branches namely: classical physics and modern physics. This book focuses on mechanics, which is a branch of classical physics. Other branches of classical physics are: light and optics, sound, electromagnetism, and thermodynamics. Mechanics is the science of motion of objects and is the core of classical physics. On the other hand, modern branches of physics include theories that have been developed during the past twentieth century. Two main theories are the theory of relativity and the theory of quantum mechanics. Modern physics explains many physical phenomena that cannot be explained by classical physics.
1.2 The SI Units
A physical quantity is a quantitative description of a physical phenomenon. For a precise description, one has to measure the physical quantity and represent this measurement by a number. Such a measurement is made by comparing the quantity with a standard; this standard is called a unit. For example, mass is a physical quantity that refers to the quantity of matter contained in an object. The unit kilogram is one of the units used to measure mass and is defined as the mass of a specific platinum–iridium alloy cylinder, kept at the International Bureau of Weights and Measures. Therefore, when we say that a block’s mass is 300 kg, we mean that it is 300 times the mass of the cylindrical platinum–iridium alloy. All units chosen should obey certain properties such as being accurate, accessible, and should remain stable under varied environmental conditions or time.

The Meter: The distance that light travels in vacuum during a time of 1/299792458 s.

The Kilogram: The mass of a specific platinum–iridium alloy cylinder, which is kept at the International Bureau of Weights and Measures.

The Second: 9192631770 periods of the radiation from cesium133 atoms.
The SI system consists of seven base fundamental units, each representing a quantity assumed to be naturally independent
Quantity  Unit name  Unit symbol 

Length  Meter  m 
Mass  Kilogram  kg 
Time  Second  s 
Temperature  Kelvin  K 
Electric Current  Ampere  A 
Luminous Intensity  Candela  cd 
Amount of Substance  mole  mol 
Prefixes for Powers of Ten
Factor  Prefix  Symbol 

\(10^{24}\)  yocto  y 
\(10^{21}\)  zepto  z 
\(10^{18}\)  atto  a 
\(10^{15}\)  femto  f 
\(10^{12}\)  pico  p 
\(10^{9}\)  nano  n 
\(10^{6}\)  micro  \(\mu \) 
\(10^{3}\)  milli  m 
\(10^{2}\)  centi  c 
\(10^{1}\)  deci  d 
\(10^{1}\)  deka  da 
\(10^{2}\)  hecto  h 
\(10^{3}\)  kilo  k 
\(10^{6}\)  mega  M 
\(10^{9}\)  giga  G 
\(10^{12}\)  tera  T 
\(10^{15}\)  peta  P 
\(10^{18}\)  exa  E 
\(10^{21}\)  zetta  Z 
1.3 Conversion Factors

\(1 \; \mathrm {m}=39.37 \; \mathrm {i}\mathrm {n}=3.281 \; \mathrm {f}\mathrm {t}=6.214 \; \times 10^{4} \; \mathrm {m}\mathrm {i}\)

\(1 \; \mathrm {k}\mathrm {g}=10^{3} \; \mathrm {g}=0.0685\) slug \(=6.02\times 10^{26} \; \mathrm {u}\)

\( 1\displaystyle \; \mathrm {s}=1.667\times 10^{2} \; \min =2.778\times 10^{4} \; \mathrm {h}=3.169\times 10^{8} \;\) yr
Example 1.1
If a tree is measured to be 10 m long, what is its length in inches and in feet?
Solution 1.1
Example 1.2
If a volume of a room is \(32 \; \mathrm {m}^{3}\), what is the volume in cubic inches?
Solution 1.2
1.4 Dimension Analysis
The symbols used to specify the dimensions of length, mass, and time are \(\mathrm {L}, \mathrm {M}\) and \(\mathrm {T}\), respectively. Dimension analysis is a method used to check the validity of an equation and to derive correct expressions. Only the same dimensions can be added or subtracted, i.e., they obey the rules of algebra. To check the validity of an equation, the terms on both sides must have the same dimension. The dimension of a physical quantity is denoted using brackets [ ]. For example, the dimension of the volume is \([V]=\mathrm {L}^{3}\), and that of acceleration is \([\mathrm {a}]=\mathrm {L}/\mathrm {T}^{3}.\)
Example 1.3
Show that the expression \(\mathrm {v}^{2}=2ax\) is dimensionally consistent, where \(\mathrm {v}\) represents the speed, x represent the displacement, and \(\mathrm {a}\) represents the acceleration of the object.
Solution 1.3
1.5 Vectors
When exploring physical quantities in nature, it is found that some quantities can be completely described by giving a number along with its unit, such as the mass of an object or the time between two events. These quantities are called scalar quantities. It is also found that other quantities are fully described by giving a number along with its unit in addition to a specified direction, such as the force on an object. These quantities are called vector quantities.
Scalar quantities have magnitude but don’t have a direction and obey the rules of ordinary arithmetic. Some examples are mass, volume, temperature, energy, pressure, and time intervals by a letter such as m, t, E\(\ldots \), etc. Vector quantities have both magnitude and direction and obey the rules of vector algebra. Examples are displacement, force, velocity, and acceleration. Analytically, a vector is specified by a bold face letter such as \(\mathbf {A}\). This notation (as used in this book) is usually used in printed material. In handwriting, the designation \(\overrightarrow{A}\) is used. The magnitude of \(\mathbf {A}\) is written as \(\mathbf {A}\) or A in print or as \(\overrightarrow{A}\) in handwriting.
A vector is represented geometrically by an arrow PQ drawn to scale as shown in Fig. 1.1. The length and direction of the arrow represent the magnitude and direction of the vector, respectively, and is independent of the choice of coordinate system. The point \(\mathrm {P}\) is called the initial point (tail of A) and \(\mathrm {Q}\) is called the terminal point (head of A).
1.6 Vector Algebra
In this section, we will discuss how mathematical operations are applied to vectors.
1.6.1 Equality of Two Vectors
1.6.2 Addition
Example 1.4
A jogger runs from her home a distance of 0.5 km due south and then 1 km to the west. Find the magnitude and direction of her resultant displacement.
Solution 1.4
1.6.3 Negative of a Vector
1.6.4 The Zero Vector
The zero vector is a vector of zero magnitude and has no defined direction. It may result from \(\mathbf {A}=\mathbf {B}\mathbf {B}=\mathbf {0}\) or from \(\mathbf {A}=c\mathbf {B}=0\) if \(c=0.\)
1.6.5 Subtraction of Vectors
1.6.6 Multiplication of a Vector by a Scalar
1.6.7 Some Properties

\(\mathbf {A}+\mathbf {B}=\mathbf {B}+\mathbf {A}\) (Commutative law of addition). This can be seen in Fig. 1.11.

\((\mathbf {A}+\mathbf {B})+\mathbf {C}=\mathbf {A}+(\mathbf {B}+\mathbf {C}),\;\) as seen from Fig. 1.12 (Associative law of addition).

\(\mathbf {A}+\mathbf {0}=\mathbf {A}\)

\(\mathbf {A}+(\mathbf {A})=\mathbf {0}\)

\(p(q\mathbf {A})=(pq)\mathbf {A}=q(p\mathbf {A})\; \) (where p and q are scalars) (Associative law for multiplication).

\((p+q)\mathbf {A}=p\mathbf {A}+q\mathbf {A} \;\) (Distributive law).

\(p(\mathbf {A}+\mathbf {B})=p\mathbf {A}+p\mathbf {B} \;\) (Distributive law).

\( 1\mathbf {A}=\mathbf {A}, \; 0\mathbf {A}=\mathbf {0}\) (Here, the zero vector has the same direction as \(\mathbf {A}\), i.e., it can have any direction), \(\; q\mathbf {0}=\mathbf {0}\)
1.6.8 The Unit Vector
The unit vector is a vector of magnitude equal to 1, and with the same direction of \(\mathbf {A}\). For every \(\mathbf {A}\ne 0, \mathbf {a}=\mathbf {A}/\mathbf {A}\) is a unit vector.
1.6.9 The Scalar (Dot) Product
1.6.9.1 Some Properties of the Scalar Product

\(\mathbf {A}\cdot \mathbf {B}=\mathbf {B}\cdot \mathbf {A}\) (Commutative law of scalar product).

\(\mathbf {A}\cdot (\mathbf {B}+\mathbf {C})=\mathbf {A}\cdot \mathbf {B}+\mathbf {A}\cdot \mathbf {C}\) (Distributive law).

\(m(\mathbf {A}\cdot \mathbf {B})=(m\mathbf {A})\cdot \mathbf {B}=\mathbf {A}\cdot (m\mathbf {B})=(\mathbf {A}\cdot \mathbf {B})m\), where m is a scalar.
1.6.10 The Vector (Cross) Product
1.6.10.1 Some Properties

\(\mathbf {A}\cdot \mathbf {A}=A^{2}, \mathbf {0}\cdot \mathbf {A}=0\)

\(\mathbf {A}\times \mathbf {B}=\mathbf {B}\times \mathbf {A}\)

\(\mathbf {A}\times (\mathbf {B}+\mathbf {C})=\mathbf {A}\times \mathbf {B}+\mathbf {A}\times \mathbf {C}\) (Distributive law).

\((\mathbf {A}+\mathbf {B})\times \mathbf {C}=\mathbf {A}\times \mathbf {C}+\mathbf {B}\times \mathbf {C}\)

\(q(\mathbf {A}\times \mathbf {B})=(q\mathbf {A})\times \mathbf {B}=\mathbf {A}\times (q\mathbf {B})=(\mathbf {A}\times \mathbf {B})q\), where q is a scalar.

\(\mathbf {A}\times \mathbf {B}=\) The area of a parallelogram that has sides A and \(\mathrm {B}\) as shown in Fig. 1.15.
1.7 Coordinate Systems
1.8 Vectors in Terms of Components
1.8.1 Rectangular Unit Vectors
1.8.2 Component Method
Suppose we have \(\mathbf {A}=A_{x}\mathbf {i}+A_{y}\mathbf {j}\) and \(\mathbf {B}=B_{x}\mathbf {i}+B_{y}\mathbf {j}\)
1.8.2.1 Addition
Example 1.5
A truck travels northwest a distance of 30 km, and then 50 km at \(30^{\mathrm {o}}\) north of east, and finally travels a distance of 20 km due south. Determine both graphically and analytically the magnitude and direction of the resultant displacement of the truck from its starting point.
Solution 1.5
1.8.2.2 Subtraction
1.8.2.3 Scalar Product
1.8.2.4 The Angle Between Two Vectors
Example 1.6
Two vectors \(\mathbf {A}\) and \(\mathrm {B}\) are given by \(\mathbf {A}=\mathbf {i}+5\mathbf {j}7\mathbf {k}\) and \(\mathbf {B}=6\mathbf {i}2\mathbf {j}+3\mathbf {k}\). Find the angle between them.
Solution 1.6
1.8.2.5 Perpendicular and Parallel Vectors
1.8.2.6 Vector Product
Example 1.7
Two vectors \(\mathbf {A}\) and \(\mathbf {B}\) are given by \(\mathbf {A}=\mathbf {i}+3\mathbf {j}\) and \(\mathbf {B}=2\mathbf {i}+\mathbf {j}\). Find: (a) the sum of \(\mathbf {A}\) and \(\mathbf {B},\cdot (\mathbf {b})\mathbf {B}\) and \(3\mathbf {A},\cdot (\mathbf {c})\mathbf {A}\cdot \mathbf {B}\) and \(\mathbf {A}\times \mathbf {B}\).
Solution 1.7
Example 1.8
Find a vector of magnitude 1 that is perpendicular to each of the vectors \(\mathbf {A}= 5\mathbf {i}+\mathbf {j}3\mathbf {k}\) and \(\mathbf {B}=3\mathbf {i}+7\mathbf {j}2\mathbf {k}.\)
Solution 1.8
Example 1.9
Given that \(\mathbf {A}=2\mathbf {i}3\mathbf {j}\mathbf {k}, \mathbf {B}=3\mathbf {i}\mathbf {j}\) and \(\mathbf {C}=\mathbf {j}4\mathbf {k}\), find (a) \(\mathbf {A}\times \mathbf {B}\) (b)\((\mathbf {A}\times \mathbf {B})\times \mathbf {C}\) (c) \(\mathbf {A}\cdot (\mathbf {B}\times \mathbf {C})\).
Solution 1.9
Example 1.10
Using vectors method, find the area of a triangle if the coordinates of its three vertices are \(\mathrm {A}(2,1,3)\) , \(\mathrm {B}(2,5,7)\) , \(\mathrm {C}(1,4,2)\) .
Solution 1.10
1.8.2.7 Triple Product
Scalar Triple Product
where \(\mathbf {A}=A_{x}\mathbf {i}+A_{y}\mathbf {j}+A_{\mathrm {z}}{\mathbf {k}}, \mathbf {B}=B_{x}\mathbf {i}+B_{y}\mathbf {j}+B_{\mathrm {z}}{\mathbf {k}}\), and \(\mathbf {C}=C_{x}\mathbf {i}+C_{y}\mathbf {j}+C_{\mathrm {z}}{\mathbf {k}}.\) Furthermore, the triple scalar product is equal to the volume of a parallepiped with sides \(\mathbf {A}, \mathbf {B}\), and \(\mathbf {C}\) as shown in Fig. 1.26. Because any edges can be used, the triple scalar product can be written as \(\mathbf {A} \cdot (\mathbf {B}\times \mathbf {C})\) or as \(\mathbf {A}\cdot (\mathbf {C}\times \mathbf {B})\) . These products are positive and negative for a righthanded coordinate system respectively. Therefore, there are 6 equal triple scalar products or 12 if you include the terms of the form \((\mathbf {B}\times \mathbf {C})\cdot \mathbf {A}\) . A. Three of these six products are positive and the rest are negative. By expanding the determinant, you can prove that
\(\mathbf {A}\cdot (\mathbf {B}\times \mathbf {C})=\mathbf {B}\cdot (\mathbf {C}\times \mathbf {A})=\mathbf {C}\cdot (\mathbf {A}\times \mathbf {B})=\mathbf {A}\cdot (\mathbf {C}\times \mathbf {B})=\mathbf {B}\cdot (\mathbf {A}\times \mathbf {C})=\mathbf {C}\cdot (\mathbf {B}\times \mathbf {A})\)
Vector Triple Product
Example 1.11
Given that \(\mathbf {A}=A_{x}\mathbf {i},\mathbf {B}=B_{x}\mathbf {i}+B_{\mathrm {z}}\mathbf {k}\), and \(\mathbf {C}=C_{y}\mathbf {j}\), show that the identity \(\mathbf {A}\times (\mathbf {B}\times \mathbf {C})=(\mathbf {A}\cdot \mathrm {C})\mathbf {B}(\mathbf {A}\cdot \mathbf {B})\mathbf {C}\) is correct.
Solution 1.11
1.9 Derivatives of Vectors
1.9.1 Some Rules
Example 1.12
Two vectors \(\mathbf {r}_{1}\) and \(\mathbf {r}_{2}\) are given by \(\mathbf {r}_{1}=2t^{2}\mathbf {i}+\cos t\mathbf {j}+4\mathbf {k}\) and \(\mathbf {r}_{2}=\sin t\mathbf {i}+\cos t \mathbf {k}\), find at \(t=0\) (a)\(\displaystyle \frac{d^{2}\mathbf {r}_{1}}{dt^{2}}\) and \((\displaystyle \mathrm {b})\frac{d(\mathbf {r}_{1}\cdot \mathbf {r}_{2})}{dt}.\)
Solution 1.12
1.9.2 Gradient, Divergence, and Curl
If \(\mathbf {A}=\mathbf {A}(x,\ y,\ z)\) is a vector function of x, y, and z then \(\mathbf {A}(x,\ y,\ z)\) is called a vector field. Similarly, the scalar function \(\phi (x,\ y,\ z)\) is called a scalar field.
1.9.2.1 Del
1.9.2.2 Gradient
1.9.2.3 Divergence
1.9.2.4 Curl
1.9.2.5 Some Identities

\(\text {divcurl}\mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A})=0\).

\(\text {curlgrad}\phi =\nabla \times (\nabla \phi )=\mathbf {0}\).
Example 1.13
A vector field A and a scalar field B are given by \(\mathbf {A}=3xy\mathbf {i}+(2y^{2}x)\mathbf {j}\) and \(B=3x^{2}y\), Find at the point (−1,1)(a) \(\nabla \cdot \mathbf {A}\) (b) \(\nabla \times \mathbf {A}\) (c) \(\nabla \mathrm {B}\).
Solution 1.13
1.10 Integrals of Vectors
1.10.1 Line Integrals
1.10.2 Independence of Path
Example 1.14
A force field is given by \(\mathbf {F}=(4xy^{2}+z^{2})\mathbf {i}+(4yx^{2})\mathbf {j}+(2xz1)\mathbf {k}\)
(a) Show that \(\nabla \times \mathbf {F}\),
(b) Find a scalar function \(\phi \) such that \(\mathbf {F}=\nabla \phi .\)
Solution 1.14
Example 1.15
A vector \(\mathbf {F}\) is given by \(\mathbf {F}=3x^{2}y\mathbf {i}(4y+x)\mathbf {j}\). Compute \(\displaystyle \int _{c}\mathbf {F}\cdot d\mathbf {r}\) along each of the following paths:
(a) The straight lines from (0, 0) to (0, 1) and then to (1, 1).
(b) Along the straight line \(y=x.\) (c) Along the curve \(x=t, y=t^{2}.\)
Solution 1.15
Example 1.16
If a vector \(\mathbf {A}\) is given by \(\mathbf {A}=xy\mathrm {i}x^{2}\mathrm {j}\), find the line integral \(\displaystyle \int _{C}\mathbf {A}\cdot d\mathbf {r}\) along the circular arc shown in Fig. 1.28.
Solution 1.16
 1.
Check if the relation \(v=\sqrt{2GM_{E}/R_{E}}\) is dimensionally correct, where v represents the escape speed of a body, \(M_{E}\) and \(R_{E}\) are the mass and radius of the earth, respectively, and G is the universal gravitational constant.
 2.
If the speed of a car is 180 \(\mathrm {k}\mathrm {m}/\mathrm {h}\), find its speed in \(\mathrm {m}/\mathrm {s}.\)
 3.
How many micrometers are there in an area of 3 \(\mathrm {k}\mathrm {m}^{2}.\)
 4.
Figure 1.29 shows vectors \(\mathbf {A}, \mathbf {B}, \mathbf {C}\), and D. Find graphically the following vectors (a) \(\mathbf {A}+2\mathbf {B}\mathbf {C}(\mathbf {b})2(\mathbf {A}\mathbf {B})+\mathbf {C}2\mathbf {D}(\mathbf {c})\) show that \((\mathbf {A}+\mathbf {B})+\mathbf {C}= \mathbf {A}+(\mathbf {B}+\mathbf {C})\) .
 5.
A car travels a distance of 1 km due east and then a distance of 0.5 km north of east. Find the magnitude and direction of the resultant displacement of the car using the algebraic method.
 6.
Prove that \(\mathbf {A}\cdot (\mathbf {B}+\mathbf {C})=\mathbf {A}\cdot \mathbf {B}+\mathbf {A}\cdot \mathbf {C}\).
 7.
A parallelogram has sides \(\mathbf {A}\) and \(\mathbf {B}\). Prove that its area is equal to \(\mathbf {A}\times \mathbf {B}.\)
 8.
If \(\mathbf {A}=2\mathbf {i}3\mathbf {j}+4\mathbf {k}\) and \(\mathbf {B}=\mathbf {i}+5\mathbf {j}2\mathbf {k}\), find (a) \(\mathbf {A}2\mathbf {B}\)(b)\(\mathbf {A}\times \mathbf {B}\) (c)\(\mathbf {A}\cdot \mathbf {B}\) (d) the length of \(\mathbf {A}\) and the length of \(\mathbf {B}\)(e) the angle between \(\mathbf {A}\) and \(\mathbf {B}\)(f) the scalar projection of \(\mathbf {A}\) on \(\mathbf {B}\) and the scalar projection of \(\mathbf {B}\) on \(\mathbf {A}\).
 9.
Show that \(\mathbf {A}\) is perpendicular to \(\mathbf {B}\) if \(\mathbf {A}+\mathbf {B}=\mathbf {A}\mathbf {B}.\)
 10.
Given that \(\mathbf {A}=2\mathbf {i}+\mathbf {j}+\mathbf {k}, \mathrm {B}=\mathbf {i}+3\mathbf {j}5\mathbf {k}\) and \(\mathbf {C}=6\mathbf {i}+3\mathbf {j}+3\mathbf {k}\), determine which vectors are perpendicular and which are parallel.
 11.
Use the vectors \(\mathbf {A}=\cos \theta \mathbf {i}+\sin \theta \mathbf {j}\) and \(\mathbf {B}=\cos \phi \mathbf {i}\sin \phi \mathbf {j}\) to prove that \(\cos (\theta +\phi )=\cos \theta \cos \phi \sin \theta \sin \phi .\)
 12.
If \(\mathbf {A}=5x^{2}y\mathbf {i}+yz\mathbf {j}3x^{2}z^{2}\mathbf {k}, \mathbf {B}=7y^{3}z\mathbf {i}2zx\mathbf {j}+xz^{2}y\mathbf {k}\) and \(\phi (x,\ y,\ z)= 2z^{2}y\), find at (−1,1,1)(a)\(\partial (\phi \mathbf {A})/\partial x\)(b)\(\partial ^{2}(\mathbf {A}\times \mathbf {B})/\partial z\partial y\)(c)\(\nabla \phi (\mathrm {d})\nabla \times (\phi \mathbf {A})\) .
 13.
Evaluate \(\nabla \times (r^{2}\mathbf {r})\) where \(\mathbf {r}=x\mathbf {i}+y\mathbf {j}z\mathbf {k}\) and \(r=\mathbf {r}.\)
 14.
If \(\mathbf {r}=A\cos \omega t\mathbf {i}+A\sin \omega t\mathbf {j}\), show that \(d^{2}\mathbf {r}/dt^{2}+\omega ^{2}\mathbf {r}=0.\)
 15.
A force field is given by \(\mathbf {F}=kx\mathbf {i}ky\mathbf {j}\), find (a) \(\nabla \times \mathbf {F}\) (b) a scalar field \(\phi \) such that \(\mathbf {F}=\nabla \phi (\mathrm {c})\) Calculate the line integral along the straight lines from (0, 0) to (1, 0) to (1, 1) and from (0, 0) to (0, 1) to (1, 1). Is the line integral independent of path?
Copyright information
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.