Chapter 1

Vectors and Scalars

1.1 Introduction
Mechanics is a physical science which deals with the state of rest or motion of rigid bodies
under the action of forces. It is divided into three parts: mechanics of rigid bodies, mechanics
of deformable bodies, and mechanics of fluids. Thus it can be inferred that Mechanics is a
physical science which deals with the external effects of force on rigid bodies. Mechanics of
rigid bodies is divided into two parts:

Statics and Dynamics

Statics: deals with the equilibrium of rigid bodies under the action of forces.

Dynamics: deals with the motion of rigid bodies caused by unbalanced force acting on them.
Dynamics is further subdivided into two parts:

 Kinematics: dealing with geometry of motion of bodies without reference to the forces
causing the motion, and
 Kinetics: deals with motion of bodies in relation to the forces causing the motion.

Basic Concepts:
The concepts and definitions of Space, Time, Mass, Force, Particle and Rigid body are basic
to the study of mechanics.
In this course, the bodies are assumed to be rigid such that whatever load applied, they don’t
deform or change shape. But translation or rotation may exist.
The loads are assumed to cause only external movement, not internal. In reality, the bodies
may deform. But the changes in shapes are assumed to be minimal and insignificant to affect
the condition of equilibrium (stability) or motion of the structure under load.
When we deal Statics/Mechanics of rigid bodies under equilibrium condition, we can
represent the body or system under a load by a particle or centerline. Thus, the general
response in terms of other load of the bodies can be spotted easily.

Fundamental Principles
The three laws of Newton are of importance while studying mechanics:

First Law: A particle remains at rest or continues to move in a straight line with uniform
velocity if there is no unbalanced force on it.

Second Law: The acceleration of a particle is proportional to the resultant force acting on it
and is in the direction of this force.
F=mxa
Third Law: The forces of action and reaction between interacting bodies are equal in
magnitude, opposite in direction, and collinear.
The first and third laws have of great importance for Statics whereas the second one is basic
for dynamics of Mechanics.
Another important law for mechanics is the Law of gravitation by Newton, as it usual to
compute the weight of bodies. Accordingly:

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1.2 Scalars and Vectors
1.2.1Definition and properties
After generally understanding quantities as Fundamental or Derived, we shall also treat them
as either Scalars or Vectors.

Scalar quantities: - are physical quantities that can be completely described (measured) by
their magnitude alone. These quantities do not need a direction to point out their application
(Just a value to quantify their measurability). They only need the magnitude and the unit of
measurement to fully describe them.
E.g. Time[s], Mass [Kg], Area [m2], Volume [m3], Density [Kg/m3], Distance [m], etc.

Vector quantities: - Like Scalar quantities, Vector quantities need a magnitude. But in
addition, they have a direction, and sometimes point of application for their complete
description. Vectors are represented by short arrows on top of the letters designating them.
E.g. Force [N, Kg.m/s2], Velocity [m/s], Acceleration [m/s2], Momentum [N.s, kg.m/s], etc.

1.2.2 Types of Vectors

Generally vectors fall into the following three basic classifications:
Free Vectors: are vectors whose action in space is not confined or associated with a unique
line in space; hence they are ‘free’ in space.
E.g. Displacement, Velocity, Acceleration, Couples, etc.

Sliding Vectors: are vectors for which a unique line in space along the action of the quantity
must be maintained.
E.g. Force acting on rigid bodies.

NB: From the above we can see that a force can be applied anywhere along its line of action
on a rigid body without altering its external effect on the body. This principle is known as
Principle of Transmissibility.

Fixed Vectors: are vectors for which a unique and well-defined point of application is
specified to have the same external effect.
E.g. Force acting on non-rigid (deformable) bodies.

1.2.3 Representation of Vectors

A) Graphical representation
Graphically, a vector is represented by a directed line segment headed by an arrow. The
length of the line segment is equal to the magnitude of the vector to some predetermined
scale and the arrow indicates the direction of the vector.

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NB: The direction of the vector may be measured by an angle υ from some known reference
direction.

B) Algebraic (arithmetic) representation

Algebraically a vector is represented by the components of the vector along the three
dimensions.
E.g.:

Properties of vectors
Equality of vectors:
 Two free vectors are said to be equal if and only if they have the same magnitude and
direction.

The Negative of a vector:

 Is a vector which has equal magnitude to a given vector but opposite in direction.
Null vector:
 It is a vector of zero magnitude. A null vector has an arbitrary direction.
Unit vector:
 any vector whose magnitude is unity.
A unit vector along the direction of a certain vector, say vector A (denoted by uA) can then be
found by dividing vector A by its magnitude.

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Generally, any two or more vectors can be aligned in different manner. But they may be:
 Collinear-Having the same line of action.
 Coplanar- Lying in the same plane.
 Concurrent- Passing through a common point.

1.3 Operations with Vectors

Scalar quantities are operated in the same way as numbers are operated. But vectors are not
and have the following rules:

1.3.1 Vector Addition /or Composition of Vectors/

Composition of vectors is the process of adding two or more vectors to get a single vector, a
Resultant, which has the same external effect as the combined effect of individual vectors on
the rigid body they act. There are different techniques of adding vectors

A) Graphical Method
I. The parallelogram law
The law states, “if A and B are two free vectors drawn on scale, the resultant (the equivalent
vector) of the vectors can be found by drawing a parallelogram having sides of these vectors,
and the resultant will be the diagonal starting from the tails of both vectors and ending at the
heads of both vectors.”

Once the parallelogram is drawn to scale, the magnitude of the resultant can be found by
measuring the diagonal and converting it to magnitude by the appropriate scale. The direction
of the resultant with respect to one of the vectors can be found by measuring the angle the
diagonal makes with that vector.
Note: As we can see in the above figure.

NB. Vector subtraction is addition of the negative of one vector to the other.

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II. The Triangle rule
The Triangle rule is a corollary to the parallelogram axiom and it is fit to be applied to more
than two vectors at once. It states “If the two vectors, which are drawn on scale, are placed tip
(head) to tail, their resultant will be the third side of the triangle which has tail at the tail of
the first vector and head at the head of the last.”

NB. From the Triangle rule it can easily be seen that if a system of vectors when joined head
to tail form a closed polygon, their resultant will be a null vector.

III. Analytic method.

The analytic methods are the direct applications of the above postulates and theorems in
which the resultant is found mathematically instead of measuring it from the drawings as in
the graphical method.

A. Trigonometric rules:
The resultant of two vectors can be found analytically from the parallelogram rule by
applying the cosine and the sine rules.
Consider the following parallelogram. And let υ be the angle between the two vectors

Consider triangle ABC, from cosine law,

This is the magnitude of the RESULTANAT of the two vectors,

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Decomposition of vectors:
Decomposition is the process of getting the components of a given vector along some other
different axis. Practically decomposition is the reverse of composition.

From sine law then,

B. Component method of vector addition

This is the most efficient method of vector addition, especially when the number of vectors to
be added is large. In this method first the components of each vector along a convenient axis
will be calculated. The sum of the components of each vector along each axis will be equal to
the components of their resultant along the respective axes. Once the components of the
resultant are found, the resultant can be found by parallelogram rule as discussed above.

1.4 Vector Multiplication: Dot and Cross products

1.4.1 Multiplication of vectors by scalars

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Multiplication of vectors by scalars obeys the following rules:
i. Scalars are distributive over vectors.

ii. Vectors are distributive over scalars.

iii. Multiplication of vectors by scalars is associative.

1.4.2 Multiplication of vector by a vector

In mechanics there are a few physical quantities that can be represented by a product of
vectors.
E.g. Work, Moment, etc
There are two types of products of vector multiplication

1.4.2 Dot Product: Scalar Product

1.4.3 Cross Product: Vector Product

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Always remember that

Moment of a Vector
The moment of a vector V about any point O is given by:

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