Geotechnical Engineering: Understanding the Foundation of Infrastructure

 

Scalars and Vectors

All the measurable physical quantities can be divided into three class’s namely

(i)              Scalar quantity,

(ii)            Vector quantity

 

 Scalar quantities: Those physical quantities which have magnitudes but not direction are called scalars. Mass, length, distance covered, time, density, work, specific heat, temperature, charge, current and pressure etc. are the examples of scalar quantities. A scalar can be completely described by a numerical value representing its magnitude. A scalar may be positive or negative. They can be added, subtracted, multiplied and divided according to the ordinary rules of algebra.

 

Vectors quantities: Those physical quantities which possess both magnitude and direction also they can obey the commutative laws of vector addition are called vectors. Displacement, velocity, acceleration, force, momentum, gravitational field, electric field, current density etc. are example of vector quantities.

 

Composition of Vectors

The process of adding two or more vectors or finding the resultant of these vectors is called composition of vectors.  For the reason that vectors possess direction in addition to their magnitudes, they cannot be added by simple laws of algebra applicable to scalars.

 

Some extra: 

Scalar and Vector

In physics and mathematics, the words scalar and vectors are commonly used to describe some quantities. Many science students have heard these words many times in the course of their studies but still most might not completely understand its meaning and what they emphasize.

Mathematical quantities are used in different fields to describe certain phenomena. When it comes to moving objects or in mechanics, there are two types of quantities used to describe this object. They are known as scalar and vector.

We will move further to define the meaning of these two quantities and some of the examples of vector and scalar quantities.

Scalars

Scalars are mathematical quantities used to describe objects of one dimensional quantity. In a simple term, they are quantities that are described in numerical terms alone. Scalar quantities have only magnitude.

Example of Scalar Quantities

It is important to list some of the scalar quantities so that it will help your understand while there are called scalar quantities.

Temperature

Time

Speed

Mass

Vectors

Unlike scalar quantities, vector quantities are quantities that can be described by magnitude and direction. This implies that vector quantities are multi-dimensional quantities.

Example of Scalar Quantities

The list below highlights some examples of vector quantities.

Velocity

Acceleration

Force

Vector Representation

Vector representation is a way to represent and fully visualize how to manipulate vectors in physics. There are different ways in which vectors can be represented in physics. The simple and commonly used is the graphical representation of vectors.

When a vector is represented graphically, its magnitude is represented by the length of an arrow and its direction is represented by the direction of the arrow.The direction of a vector is often expressed as a counter-clockwise angle of rotation of that vector from due east. The graphical representations of vectors use the concept of tip and tail. The tip has an arrow which indicates the direction of the vector.

Beside graphical representation of vectors, vectors can be expressed in terms of their components. Being able to translate between the two representations is an essential skill in physics. The magnitude of a vector is its length. The direction is usually given in terms of some angle.

Vector Addition

Vector addition involves the summation of two or more vectors to get the resultant vector. The addition of vectors is one of the many operations that can be performed with vectors. To understand the in-depth of vector addition, let us move back to the Newton’s first law of motion which stated that an object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

This implies that the net force experienced by an object is equal to the sum of the resultant forces that act on the object. In some situation, we make use of an unbalance force when we are talking about the forces that act on objects. In its simplicity, unbalance force is when that forces that act on an object in opposite direction is not equal to zero when they cancel each other.

We will go ahead to show a diagram that illustrates vector addition in its simplest form.

Another important vector addition that is worth mentioning is that Pythagoras theorem can be used to add two vector that are at right-angle to each other.