The velocity vector gives two pieces of information about a path - the direction of movement (the direction in which the velocity vector points) and the speed (the length of the velocity vector). If we normalize the velocity vector (remember normalize means create a parallel vector of length 1) then we create a vector that solely tells us about direction of movement (and how rapidly that direction is changing). This vector is called the [b][color=#ff0000]unit tangent[/color][/b] vector to the path [math]\vec{c}[/math]. In other words:[br][br][math]\vec{T}\left(t\right)=\frac{\vec{c}\,'\left(t\right)}{\left|\left|\vec{c}\,'\left(t\right)\right|\right|}[/math][br][br]In the GeoGebra applet below you can see a path traced out by a moving point with the unit tangent anchored on the moving point. In the upper window, the unit tangent is drawn in standard position (i.e. anchored at the origin). Spend some time experimenting. Consider the following questions:[br][list][*]For every differentiable path [math]\vec{c}[/math], the unit tangent (when anchored at the origin) traces a portion of the same curve. Explain.[/*][*]What is happening in the path when the unit tangent is swiveling slowly? What about when it is swiveling rapidly?[/*][*]Can you design a path whose unit tangent (when drawn in standard position) traces out an injective path?[/*][*]Can you design a path whose unit tangent (when drawn in standard position) traces out an entire circle?[/*][*]Can you design a path whose unit tangent (when drawn in standard position) does not move at all?[/*][/list]