My calculus book featured several graphing problems that involved drawing the position vector and the tangent vector at a certain point on the graph of a vector equation. Trying to trace several of these in my head, I realized that it seems like the tangent vector will trace the position vector at (at least) one point.
Is this always true (true at all), and how would one find such such a point?
The tangent vector, based at the position vector, will point in the direction in which the curve is progressing. If it actually points along the path that the position vector traces out (that is the curve in question) then the curve is a line. Or, at a minimum, is is locally close to a line. Generally, the curve does not follow the tangent line. In fact, the tendency of the curve to lifts off that line is measured by the curvature. Then the tendency of the curve to lift of the plane which contains the "osculating" circle (infinitesimal plane of motion) is measured by non-trivial torsion. You should read on the Frenet Frame (T,N,B). It will reveal all these mysteries and much more.