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# Scalars and Vectors

In today’s world, there are various mathematical quantities used in physics. Some of the examples include displacement, velocity, acceleration, speed, force etc. These quantities are either described as a scalar quantity or a vector quantity which distinguishes from each other by their specific definitions. Below we will discuss in detail about the scalar quantities and vector quantities.

## The need for vectors

Whenever the motion of an object is to be described, a certain fixed point called the origin is chosen as the reference point. In one dimensional motion, the direction corresponding to increasing position co-ordinate is chosen as positive and the opposite direction corresponding to decreasing co-ordinate as negative. If the body is moving towards the increasing position co-ordinate, its velocity is taken as positive. On the other hand, if the body is moving in the direction of the decreasing co-ordinate, its velocity is taken as negative.

when the motion of objects in a two-dimensional plane or three-dimensional space is considered, the concept of direction becomes much more important than that in one dimension. In one dimensional motion, there are only two possible directions. But, for motion in a two-dimensional plane or three-dimensional space, any number of directions are possible. The quantities like displacement, velocity, acceleration, force etc cannot be represented by magnitude alone. A complete idea of these quantities will be obtained only if their magnitudes, as well as their directions, are specified. Such quantities having both magnitude and direction are called vectors. To deal with vector quantities a new branch of mathematics called vector analysis has been developed.

## Define scalar and vector quantity

Physical quantities are classified into two: scalars and vectors.

The physical quantities which have only magnitude but no direction are called scalar quantities or scalars.
Examples: Volume, mass, time, distance, speed, density, temperature, angle, work, pressure etc.
A scalar is specified by mere number and unit, where the number represents its magnitude. A scalar may be positive or negative and they have no direction.

The physical quantities which have both magnitude and direction are called vector quantities or vectors.
Examples: Displacement, velocity, momentum, force, acceleration, torque etc.

## Difference between Scalars and Vectors

 Scalars Vectors They have only magnitude They have both magnitude and direction. They change with change in magnitude. They change with change in magnitude or with direction or both. They can be added, subtracted, multiplied or divided according to ordinary rules of the algebra. They can be added, subtracted or multiplied by using laws of vectors. Represented by ordinary letters. Represented by bold letters or with an arrow over a letter. Eg: A or $overrightarrow{A}$ $overrightarrow{A}$

## Position vector and Displacement vector

A vector representing the position of an object (point) A with respect to an origin O is called the position vector $overrightarrow{OA}$ of the object. It may be represented by an arrow with tail at O and head at A. The position vector $overrightarrow{OA}$ indicates the distance and direction of the object with respect to the chosen origin.

Let A be the position of a moving object at time t and similarly A’ be the position of a moving object at a later time t’ (Figure 2). $overrightarrow{OA}$ and $overrightarrow{OA'}$
are the position vectors at times t and t’ respectively. So the vector $overrightarrow{AA'}$
is called the displacement vector corresponding to the motion in the time t to t’. The actual path travelled by the object may not be the straight line joining A to A’.

## Representation of a vector

A vector can be represented either by a single bold letter or by a single letter with an arrowhead on it. For example, force is represented by $overrightarrow{F}$ or F. Its magnitude is represented by $|overrightarrow{F} |$, called modulus of the force vector. The modulus or magnitude of the vector is a scalar quantity.

The vector can be represented graphically or geometrically by a straight line with an arrowhead. The length of the line gives the magnitude of the vector while the arrowhead indicates its direction.

Figure 3 represents a vector $overrightarrow{A}$ = $overrightarrow{OA}$. Here OA is the magnitude of the given vector and the direction is shown by the arrow (head). The initial or starting point O of the arrow line is called the origin or tail of the vector. The point A which is at the end of the arrow line is called tip or the head or the terminus of the vector.

## Modulus of a vector

Modulus of a vector is a positive number which measures the magnitude of the vector.
For example, the modulus of a vector $overrightarrow{A}$ is represented as $|overrightarrow{A}|$ or, simply as A.

## Characteristics of a vector

The characteristics of a vector are given below:

• A vector has both magnitudes as well as direction.
• A vector does not obey the ordinary rules of Algebra.
• A vector may change with the change in magnitude or with the direction, or both.