The two vectors defined in the previous activity were either always horizontal ([math]\vec{v}_h\left(t\right)[/math]) or always vertical ([math]\vec{v}_v\left(t\right)[/math]). One appeared to change directions periodically while the other never did. The gifs below illustrate both vectors. Explain.

The sum of these two vectors is the [b][color=#ff0000]velocity vector[/color] [/b]to the path [math]\vec{c}\left(t\right)[/math]. That is, the velocity vector to the path [math]\vec{c}\left(t\right)=\left(x\left(t\right),y\left(t\right)\right)[/math] is the vector [math]\vec{c}\,'\left(t\right)=\vec{v}\left(t\right)=\left(x'\left(t\right),y'\left(t\right)\right)[/math].[br][br]With this new vocabulary word we can rewrite the definition of a differentiable and a regular:[br][list][*]A [b][color=#ff0000]differentiable path[/color][/b] is a function [math]\vec{c}:\left[a,b\right]\to\mathbb{R}^2[/math] for which [math]\vec{c}\,'\left(t\right)[/math] exists at every point in the domain. ([i]path property)[/i][/*][*]A [b][color=#ff0000]differentiable curve[/color][/b] is one for which there is a differentiable parameterization. ([i]curve property)[/i][/*][*]A [b][color=#ff0000]regular path[/color][/b] is one for which [math]\vec{c}\,'\left(t\right)[/math] exists and is never [math]\vec{0}[/math] at any point in the domain. ([i]path property)[/i][/*][/list]

In the GeoGebra applet below you can view the interactions between the vectors [math]\vec{v}_h\left(t\right),\vec{v}_v\left(t\right),\text{ and the velocty vector }\vec{c}\,'\left(t\right)=\vec{v}\left(t\right)=\vec{v}_h\left(t\right)+\vec{v}_v\left(t\right)[/math].