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3.3 Tangent vector to a curve

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

dydx=dy/dtdx/dt.{{dy}\over{dx}}={{dy/dt}\over{dx/dt}}.

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

dxdt=velocity along horizontal axis,{{dx}\over{dt}}={\hbox{velocity along horizontal axis}},
dydt=velocity along vertical axis;{{dy}\over{dt}}={\hbox{velocity along vertical axis}};

speed is the magnitude of velocity.

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