Why concept of dimension has basic importance ?

A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E} \times \vec{B}) / \mu_0$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $\mu_0$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity (ies)?

$(A)$ $\frac{\text { Energy }}{\text { charge } \times \text { current }}$

$(B)$ $\frac{\text { Force }}{\text { Length } \times \text { Time }}$

$(C)$ $\frac{\text { Energy }}{\text { Volume }}$

$(D)$ $\frac{\text { Power }}{\text { Area }}$

If Surface tension $(S)$, Moment of Inertia $(I)$ and Planck’s constant $(h)$, were to be taken as the fundamental units, the dimensional formula for linear momentum would be

Using dimensional analysis, the resistivity in terms of fundamental constants $h, m_{e}, c, e, \varepsilon_{0}$ can be expressed as

A quantity $x$ is given by $\left( IF v^{2} / WL ^{4}\right)$ in terms of moment of inertia $I,$ force $F$, velocity $v$, work $W$ and Length $L$. The dimensional formula for $x$ is same as that of

Given that $v$ is the speed, $r$ is radius and $g$ is acceleration due to gravity. Which of the following is dimensionless?