I am given the equation for the potential of an arbitrary dipole. I need to draw the electric field lines for this dipole in a plane, and also show that these lines are perpendicular to the equipotential lines. I have already derived the equation for the electric field using the gradient of the potential and mapped out the equipotential lines.
V d i p ( ⃗r ) = Constant ⃗r ·p⃗ /(r^3)
E⃗ d i p = − ∇⃗ V d i p
= (Constant) 3( ⃗r · p⃗) ⃗r /(r^5) - p⃗/(r^3)
Take p⃗ to equal a unit vector for an orthonormal basis. Such as the unit vector for x in the x, y, z coordinate system.
The Attempt at a Solution
I know that the gradient of V is always perpendicular to V, so intuitively this makes complete sense. However, I do not know how to show that a scaler quantity (V) is perpendicular to the vector equation I derived for E. I am also unsure how to map such a strange function for E into ℝ2 although obviously I know what it looks like.