# Scalars and Vectors

All the measurable physical quantities can be divided into three classes; scalars, vectors and tensors.

## Scalars

Those physical quantities which have magnitude only but no direction are called scalars. Mass, length, time, density, speed, work, etc. are the examples of scalar quantities.

A scalar can be completely described by a numerical value representing its magnitude. It may be positive or negative. They can be added, subtracted, multiplied and divided according to the ordinary rules of algebra.

## Vectors

The word vector is derived from the Latin word vehere which means ‘to carry‘. According to A.N. whitehead, “The idea of vector, i.e., of directed magnitude, is the root idea of physical science.

Vectors are dynamic concepts that appeal directly to our intuition and, also provide us with vivid illustrations of both physical and geometrical phenomena.

Those quantities which have both magnitude and direction and also they can obey the commutative laws of vector addition are called vectors. Displacement, velocity, acceleration, momentum, force, gravitational field, electric field, etc. are the examples of vector quantities.

### Notation and Representation

Generally, the vectors are denoted by bold faced type of letters known as Clarendon letters. But since the bold faced type of letter is inconvenient to indicate the vectors in writing, so we may use the following notation to represent a vector.

In mathematical expression, a vector quantity is represented by a letter or a combination of two letters with an arrow over it. And, a scalar is represented by the same letter or letters without any arrow over it.

Geometrically, a vector is represented by a directed line segment i.e. a line segment with an arrow at one end. For this, we require two points say $O$ and $P$ such that the magnitude of the vector is the length of the line segment $OP$ and its direction is from $O$ to $P$. This vector is represented by OP (Clarendon letter) or $\overrightarrow{OP}$. If $\overrightarrow{OP}$ represents force vector, then it may also represented by F or $\overrightarrow{F}$.

For the directed line segment $\overrightarrow{OP}$, the point $O$ is called the initial point or origin, and the point $P$ is called the terminal point or terminus.

In many problems, the location of the origin of a vector is immaterial. Accordingly, a vector will be represented by a directed line segment whose origin can be chosen at our convenience.

It may be pointed out that the vectors cannot be added, subtracted, multiplied, or divided as one may do in case of scalars. Since, in addition to magnitude, vectors have direction also, so they are added, subtracted and multiplied according to the vector algebra. The division of a vector by another is not defined.

## Tensors

Those physical quantities which have no specified direction, but have different values in different directions are called tensors. For example, moment of inertia, stress, strain, density, refractive index, electrical conductivity, etc. are normally scalars, but in axisotropic media, they assume different values in different directions and so become tensors.

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