3.3 Tangent vector to a curve

We suppose that the curve $C$ is smooth, so that $C$ has tangents at each point. The tangent is in the direction of the vector $({{dx}\over{dt}},{{dy}\over{dt}})$ at time $t$; in particular, a line that is tangent to $C$ at $P=(x(t),y(t))$ has gradient

If the moving point were to fly off $C$, then it would leave in the direction $({{dx}\over{dt}},{{dy}\over{dt}})$. The vector $({{dx}\over{dt}},{{dy}\over{dt}})$ is the velocity vector, which has components

speed is the magnitude of velocity.

The normal to $C$ at $P$ is the line through $P$ that is perpendicular to the tangent, hence has gradient $-dx/dy$.