BASICS OF ELECTRICITY AND ELECTRICAL CIRCUIT /ELN 113
CHAPTER 1: MATHEMATICAL TOOLS
Learning Objectives
By the end of this section, you will be able to:
• Describe vectors in two and three dimensions in terms of their components, using unit vectors along the axes.
• Distinguish between the vector components of a vector and the scalar components of a vector.
• Apply the scalar and product on vectors
• Have the knowledge on scalar field, vector field and vector flow
INTRODUCTION
Vectors are usually described in terms of their components in a coordinate system. Even in
everyday life we naturally invoke the concept of orthogonal projections in a rectangular
coordinate system. For example, if you ask someone for directions to a particular location,
you will more likely be told to go 40 km east and 30 km north than 50 km in the
direction 37° north of east.
I. Definition of a vector
There are two defining operations for vectors
I.1. Vectors addition
BASICS OF ELECTRICITY AND ELECTRICAL CIRCUIT /ELN 113
II. decomposition of a vector
Suppose we have a force F that makes an angle of 30° with the positive x axis, we want to
decompose F into x and y components.
The first thing we need to do is to represent the two components on the xy-plane. We do this
by dropping two perpendiculars from the head of F: one to the x axis, the other to the y axis.
And we join the origin of the xy-plane with the x-intercept to represent the x component of F. And
again, we join the origin with the y-intercept to represent the y component of F:
Fx and Fy are two vectors, i.e. they both have a magnitude and a direction. However, since Fx
and Fy are in the directions of the x and y axes, they are commonly expressed by the magnitude
alone, preceded by a positive or negative sign: positive when they point in the positive
directions, and negative when they point in the negative directions of the x and y axes.
In our example Fx and Fy are positive because both point in the positive directions of the x and
y axes.
BASICS OF ELECTRICITY AND ELECTRICAL CIRCUIT /ELN 113
The positive values of Fx and Fy can be found using trigonometry:
Fx = F cos 30°
Fy = F sin 30°
Let just remember that if a component is adjacent to the angle, then it is cos, otherwise it is sin.
Example: let’s determine the components Fx and Fy using the figures below:
Exercise: Calculate the x and y components of the force for the different cases below:
a) A force of 19 N is in the direction of the negative x axis.
b) A force of 114 N makes an angle of 67° with the positive x axis.
c) A force makes an angle of 221° with the positive x axis. Assuming the force has
magnitude 3.1 × 103 N.
d) A force of 4.5 × 105 N has the direction of the positive y axis.
e) A 90.0 N force makes an angle of 33° with the positive y direction.
f) A force that has magnitude 3.21 × 104 N makes an angle of −50° with the positive x
axis.
III. Dot or scalar product of vectors
The dot product of vectors is also called the scalar product of vectors. The resultant of the dot
product of the vectors is a scalar value. Dot Product of vectors is equal to the product of the
magnitudes of the two vectors, and the cosine of the angle between the two vectors. The
resultant of the dot product of two vectors lie in the same plane of the two vectors. The dot
product may be a positive real number or a negative real number.
BASICS OF ELECTRICITY AND ELECTRICAL CIRCUIT /ELN 113
Let a and b be two non-zero vectors, and θ be the included angle of the vectors. Then the scalar
product or dot product is denoted by a.b, which is defined as:
IV. Cross product of vectors
Cross Product is also called a Vector Product. Cross product is a form of vector multiplication,
performed between two vectors of different nature or kinds. When two vectors are multiplied
with each other and the product is also a vector quantity, then the resultant vector is called the
cross product of two vectors or the vector product. The resultant vector is perpendicular to the
plane containing the two given vectors.
We can understand this with an example that if we have two vectors lying in the X-Y plane,
then their cross product will give a resultant vector in the direction of the Z-axis, which is
perpendicular to the XY plane. The × symbol is used between the original vectors. The vector
product or the cross product of two vectors is shown as:
BASICS OF ELECTRICITY AND ELECTRICAL CIRCUIT /ELN 113
Find the cross product of u and v
Lecturer : Mr. Gabien THIBOU
