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Electrostatic force is given by F = qE, where F and E are both vector quantities. If the dot product of either side and the displacement vector Δs along an equipotential line is taken, the equation becomes

F⋅Δs = qE⋅Δs.

F and E are parallel; equipotential lines surround a charge radially. Since no work is done to move a charge along an equipotential line, F⋅Δs = 0. This means

qE⋅Δs = 0,

but q, E, and Δs are assumed to be nonzero. The only way to make the equation zero is to make E perpendicular to Δs (i.e., cosθ = 0) in all cases.

I would really appreciate a better explanation and pointing out of any errors in my understanding.