You may have noticed that when I animate a path by tracing the movement of a point, the dots along the path are not evenly spaced apart. Unsurprisingly this is related to the way the vector [math]\vec{c}\left(t\right)[/math] is changing. In the GeoGebra applet below, a path is animated on the left screen while the length of the velocity vector [math]\vec{c}'\left(t\right)[/math] is tracked on the right screen.

The length of the velocity vector is called the [b][color=#ff0000]speed[/color][/b] of the path [math]\vec{c}\left(t\right)[/math]. That is:[br]The [b][color=#ff0000]speed[/color][/b] of the path [math]\vec{c}\left(t\right)[/math] is the scalar quantity [math]\left|\left|\vec{c}'\left(t\right)\right|\right|[/math].[br][br]Why do you think we chose the word "speed" to describe this scalar quantity?