the principal normal vector

Drive along a curving road and you feel pushed sideways — that sideways pull is toward the inside of the bend. The principal normal vector is the unit arrow pointing in exactly that sideways direction: perpendicular to your direction of travel, lying in the plane in which the curve is momentarily turning, and aimed toward the center of the turn. It answers the question 'which way is the curve bending right now?'.

Here is how it arises. Work with the arc-length parametrization so the unit tangent is T = r'(s) and |T| = 1. Because T has constant length 1, differentiating the relation T . T = 1 gives 2 T . T' = 0, so T'(s) is always perpendicular to T — the tangent can only turn sideways, never lengthen. The length of T'(s) is the curvature kappa, measuring how fast the direction is changing. Provided kappa is not zero, the principal normal is the unit vector in the direction of this turning: N = T'(s) / |T'(s)| = T'(s) / kappa. By construction N is perpendicular to T and points to the concave side of the curve, toward the center of the osculating circle. Compactly, T'(s) = kappa N, which says the tangent turns toward N at rate kappa.

N is the second member of the moving frame, and together with T it spans the osculating plane, the plane that best hugs the curve at that point. Where it matters: the acceleration of any motion splits into a part along T (speeding up or slowing down) and a part along N (changing direction), and that second part is centripetal, always pointing toward the inside of the bend — which is why N is sometimes called the centripetal direction. Two honest cautions. First, N is undefined at points where kappa = 0, for instance along a straight stretch, because there the tangent is not turning and 'the inside of the bend' has no meaning. Second, for a plane curve people often prefer a normal chosen by a fixed left-or-right rule (giving signed curvature), which can point opposite to this principal normal; the principal normal here always points toward the concave side.