Question

Define dimension of a physical quantity.

Answer 1

Dimension of a physical quantity is the power to which the fundamental units must be raised to, in order to represent it.
Mass, length, time, temperature, electric current, luminous intensity and amount of substance are the fundamental quantities. Physical quantities can be expressed in terms of these fundamental quantities. These seven quantities are the seven dimensions of the physical world.
The dimension of mass is denoted by $\left[ M \right]$, dimension of length is denoted by $\left[ L \right]$, dimension of time is denoted by $\left[ T \right]$, dimension of temperature is denoted by $\left[ K \right]$, dimension of electric current is denoted by $\left[ I \right]$, dimension of luminous intensity is denoted by $\left[ {cd} \right]$ and amount of substance is denoted by $\left[ {mol} \right]$.

Complete step by step answer:
There are seven fundamental quantities. They are mass, length, time, temperature, electric current, luminous intensity and amount of substance. Physical quantities can be expressed in terms of these fundamental quantities. These seven quantities are the seven dimensions of the physical world.
The dimension of mass is denoted by $\left[ M \right]$, dimension of length is denoted by $\left[ L \right]$, dimension of time is denoted by $\left[ T \right]$, dimension of temperature is denoted by $\left[ K \right]$, dimension of electric current is denoted by $\left[ I \right]$, dimension of luminous intensity is denoted by $\left[ {cd} \right]$ and amount of substance is denoted by $\left[ {mol} \right]$
These letters specify only the unit and not its magnitude.
Dimension of a physical quantity can be said as the power to which these fundamental units must be raised in order to represent it.
Let us consider an example. Area of a rectangle is $6\,{m^2}$. We need to find the dimension of this area.
We know that the area of a rectangle is length multiplied by breadth. Both length and breadth have unit $m$ Thus, their product has a unit ${m^2}$. Dimension of length is written as $\left[ L \right]$. It is meter square here so the dimension will be $\left[ {{L^2}} \right]$.
Let us now see how we can find the dimension of density.
We know density is mass divided by volume. Dimension of mass is $\left[ M \right]$. Volume is a product of three lengths. Thus, the dimension of volume will be $\left[ {{L^3}} \right]$
So, the dimension of density can be written as
${\text{density}} = \dfrac{{{\text{mass}}}}{{{\text{volume}}}} \\\ = \dfrac{{\left[ M \right]}}{{\left[ {{L^3}} \right]}} \\\ = \left[ {M{L^{ - 3}}} \right] \\\$

Note: The dimension of a physical quantity will be the same as that of the dimension of its unit.
There are some cases where some quantities are dimensionless even when they have a unit. For example, angle is a dimensionless quantity.