I studied the equation for current density $$I=\vec{J}\cdot\vec{A}$$ but nowhere is the significance of the direction of current density mentioned.
Also, I want to know why we use dot product instead of cross product, though I know we use it because current is a scalar but still that doesn't explain it any better.
Current density $\mathbf{J}$ is the rate of flow of charge per unit area I.e the flux of charge through a surface with unit area. This prompts the equation: \begin{equation} \mathbf{J} = n q \mathbf{v} \end{equation} Where $n$ is the number density of charges, $q$ is the charge and $\mathbf{v}$ is the velocity vector. If $\mathbf{J}$ is antiparallel to the velocity, that means the current is made up of negative charges. If the two vectors are parallel, the current constitutes the flow of positive charges. To calculate the current through an area $A$ you need to use a dot product, since the current density $\mathbf{J}$ may not be perpendicular to the surface defined by the area $A$. Therefore it is natural to define a vector $\mathbf{A}$ which has magnitude $A$ and is perpendicular to the surface. This way, the dot product \begin{equation} I = \mathbf{J} \cdot \mathbf{A}= JAcos(\theta) \end{equation} describes the total charge flux per unit time, through the surface. ($\theta$ is the angle between the current density vector and the area vector)
First of all, using a cross product here is very far from the definition. If you familiarize with the concept of flux, this will be easier to understand. The current is often defined as the change of the amount of charge in a 'slice' of the conducting material at a given time, current density is a vector function that defines how much charge is flowing through at a given time, and the direction in which they are flowing. So current can really be defined as the flux of the current density through a surface, which is what that dot product implies.
This definition is especially useful for analyzing magnetic fields of a system. Since magnetic fields are created by moving charges and their direction depends on the direction of the moving charges, current density offers all the information you need to deduce the magnetic field that is created in the system.
Also note, a cross product here would output $I$ as another vector which is perpendicular to both the normal of the area and the current density. For a real world example, this would imply the current inside a wire would be perpendicular to the wire.
It is very similar to the formula used in electric flux that is , Electric flux (which is scalar) given by the Dot product of two vector,( Electric field and Area) .(Dot product of two vectors is scalar) Similarly here by dot product of Current density and Area we get Current , a Scalar.
