Vector Functions and Their Derivatives

A vector function r(t)=(x(t),y(t),z(t)) describes a curve in space. Imagine the vector starting at the origin and its tip tracing the curve. Below is the vector function r(t)=(cos t, sin t, t) and the helix it traces in space:

The derivative of the vector function r(t) is the vector function . The following two animations show how geometrically the derivative ends up being a vector tangent to the curve at that point. The example is r(t)=(1+cos t, 1+sin t, t) tracing another helix. Only part of it is shown. The derivative is computed at t=1.

The first picture shows the vectors r(1), r(1+h) and r(1+h)-r(1). The first two are position vectors and start at the origin, point to the curve. The difference of the two, shown in red, starts at the point where t=1 so at (1+ cos 1, 1+ sin 1, 1). As h approaches 0, the red vector shrinks in size and disappears. The vectors do not have arrows at their tips despite my struggle with Maple for a good amount of time. Only the tips are moving. The base points stay put.

The next picture shows the vectors r(1), r(1+h) and . The red vector is almost the derivative r’(1) when the animation ends at h=0.01. It is tangent to the curve at that moment.