Hello student,
First of all, select two arbitrary points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) on
Ax + By + Cz + D₁ = 0 and Ax + By + Cz + D₂ = 0 respectively.
Now, therefore, the vector t = <x₂-x₁, y₂-y₁, z₂-z₁> is the vector starting at P₁ and ending at P₂.
Moreover, the projection of vector t onto the normal vector n = <a, b, c> is the distance between the two planes.
This length will therefore be given by the modulus of the projection of t onto n and is therefore given as under:

Now, since the points lie on the given planes so we have
Ax₁ + By₁ + Cz₁ = -D₁ and Ax₂ + By₂ + Cz₂ = -D₂.
Hence, using these the above expression reduces to

This is the required formula.