the binormal vector
A curve in space, like a helical spring or a roller-coaster track, does not just bend within a flat plane — it can also twist out of that plane, lifting like a corkscrew. To measure that twisting you first need a reference: a direction perpendicular to the plane the curve is momentarily turning in. The binormal vector supplies exactly this. It is the normal to the osculating plane, the unit arrow that sticks straight out of the flat disk the curve is curling around at that instant.
It is built from the two frame vectors you already have, by the cross product: B = T x N. Because T and N are perpendicular unit vectors, their cross product B is automatically a unit vector perpendicular to both, and the three vectors T, N, B form a right-handed orthonormal triple, like the (x, y, z) axes carried along with the moving point. The plane spanned by T and N is the osculating plane; B is its normal. As you travel along the curve, the way B tilts measures how the osculating plane itself is rotating in space — and the rate of that tilt is the torsion. In fact B'(s) = -tau N, so the binormal changes only in the N direction, at a rate equal to the torsion tau.
Where it matters: B is what makes space-curve geometry richer than plane-curve geometry. For a flat (planar) curve the osculating plane never changes, so B is constant and the torsion is zero everywhere — twisting is exactly what B detects. The honest caveats inherited from N apply here too: B is built from N, so it is undefined wherever the curvature is zero (no osculating plane is singled out there). And like the rest of the frame, B depends on orientation; reversing the direction of travel can flip the handedness bookkeeping, so one must keep the convention B = T x N fixed and track it consistently.
A circular helix r(t) = (cos t, sin t, t) climbs as it turns. Its osculating plane tilts steadily, and the binormal B makes a constant angle with the vertical axis, slowly precessing around it. The torsion tau is a nonzero constant here, which is exactly the statement 'this curve twists out of any single plane at a steady rate'.
B = T x N completes a right-handed frame; how B tilts is the torsion.
For a plane curve the binormal is constant and the torsion is zero — B only becomes interesting in three dimensions. Like N, the binormal is undefined where the curvature vanishes, since no osculating plane is determined there.