# Vector And Scalar Quantities

In mathematics and physics, we have physical quantities which can be categorized in two ways, namely it’s a scalar quantity or a vector quantity. Let us now see what vector and scalar is?

The physical quantities which have only magnitude are known as scalar quantities. For example, if I say that the height of a tower is $15 meters$ then height of the tower is a scalar quantity as it needs only the magnitude of height to define itself. Let’s take another example, suppose the time taken to complete a piece of work is $3 hours$, then in this case also to describe time just need the magnitude i.e.  $3 hours$.

Other examples of scalar quantities are mass, speed, distance, time, energy, density, volume, temperature, distance, work and so on. In simple language scalar is a measure of a quantity.

Coming to vectors, the physical quantities for which both magnitude and direction are defined distinctly are known as vector quantities. For example, a boy is riding a bike with a velocity of $30km/hr$ in north-east direction. Then, as we see for defining the velocity, we need two things, i.e. the magnitude of velocity and its direction. Therefore, it represents a vector quantity. The quantities such as displacement, acceleration, force are vectors.

Representation of a Vector:

Let us have a look at the line segment drawn below.A vector quantity always has a starting point and an end point. The two end points of the given line segment are distinguishable as and. It represents a directed line segment

The directed line segment with an initial point $A$and terminal point $B$ is symbolically denoted as $AB$ in bold or as $\overrightarrow{AB}$ and the length $a$ of the vector represents its magnitude which is denoted by $|AB|$. Instead of using double letter notation we can use a single letter notation to represent a vector as $a, b, c$ and the letters $a, b, c$ can denote their magnitudes. As it is difficult to write letters in bold we use a bar above the letters to represent vectors as $\overline{a}$.

Therefore, if $\overrightarrow{AB} = a$ then$|\overrightarrow{AB}| = a$ where $|\overrightarrow{AB}|$ indicates the magnitude of the vector. Also the magnitude is called as the modulus.

Now we are familiar with what are vectors and scalars. Now if somebody asks if acceleration is a vector or a scalar, we can easily tell that it’s a vector because it has direction as well as magnitude. Similarly, when asked if distance is a vector or scalar, it is quite evident that as distance has only magnitude, it is a scalar quantity.

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#### Practise This Question

If matrix A=[aij]3×3,matrix B=[bij]3×3 where aij+aji=0 and bijbji=0,then |A4.B3| is