Introduction

In day-to-day life physical quantities like temperature, work and distance can be represented wholly by their magnitude alone. The relation between these physical quantities can be explained with simple laws of arithmetic. Such quantities are known as scalars.

To represent certain physical quantities like displacement, acceleration and force, direction is also necessary along with the magnitude. The physical quantities, which have magnitude and direction and obey vector laws are known as vectors. The vectors and vector algebra help in analyzing and solving physical problems independent of any coordinate system. Some basic concepts to vectors are discussed in this chapter.

Objectives

In this chapter, we are going to learn about
• Scalars and vectors
• Position and displacement vectors
• Equality of vectors
• Multiplication of vectors by real numbers
• Addition and subtraction of vectors-graphical method
• Resolution of vectors
• Motion in a plane
• Position vector and displacement
• Velocity
• Acceleration
• Motion in a plane with constant acceleration
• Relative velocity in two dimensions
• Projectile motion
• Uniform circular motion

Scalars and Vectors

All those quantities, which can be measured, are known as physical quantities. These quantities can be broadly classified into two categories â€“ scalar quantities and vector quantities.

Scalar quantities are those physical quantities, which are characterized by magnitude only.

These directionless quantities are briefly called scalars. These obey the ordinary laws of Algebra. A scalar quantity is completely specified by merely stating a number. A few examples of scalars are volume, mass, speed, density, number of moles, angular frequency, temperature, pressure, time, power, total path length, energy, gravitational potential, coefficient of friction, charge and specific heat.

Vector quantities are those physical quantities, which are characterized by both magnitude and direction.

These quantities are briefly called vectors. A vector is specified not by merely stating a number but a direction as well. Since the concept of vectors involves the idea of direction, vectors do not follow the ordinary laws of Algebra. A few examples of vectors are: displacement, velocity, angular velocity, acceleration, impulse, force, angular momentum, current, linear momentum, electric field, magnetic moment and magnetic field.

Representation of a Vector

A Vector is represented by a line with an arrow head. In figure a vector is represented by a directed line PQ. The length of the line gives the magnitude of the vector. The magnitude of the vector is called the modulus of the vector. The direction of the arrow represents the direction of the vector. The point P from where the arrow starts is called the tail or initial point or origin of the vector. The point Q where the arrow ends is called the tip or head or terminal point or terminus of the vector. In books, vectors are sometimes represented by bold-faced letters. As an example, a may be written as a.

General Vectors and a New Notation

A Vector quantity possesses both magnitude and direction. But any quantity which has magnitude and direction is not necessarily a vector. This is because vectors obey different laws and it is just possible that a quantity, which has both magnitude and direction, may not obey the vector laws. It is of course true that we do not come across such cases very often. But we cannot overlook this fact.

In the case of position and displacement vectors, both the tail and the tip of the arrow convey an ideal of position in space. This may not be true in the case of other vectors. As an example, let a be a particle be at any instant of time. At this instant of time, its velocity is shown to be while the force it experiences is . So, the tail-tip representation of position and displacement vectors is not relevant in this case. In this case, the tip has no direct meaning as such. As an example in the present case it will be useless to write and Instead we shall simply write and .