On the wall it is easy to translate discussing vectors into data on lengths and slopes. When Pete invented World Wallpaper he wanted to describe his trips as continuous curves on the Wallpaper tessellated with Mercator maps. Using vectors to describe a trip is exactly like telling someone the sequence of directions to take to reach a specific destination. To show why Pete's idea of World Wallpaper is impossible, we use argument by contradiction. This starts by pretending that will World Wallpaper is possible. Then, you compare trips using World Wallpaper with trips on a globe of the world. Using vectors as the essential data of a trip in both contexts allows such a comparison. Vectors work as well in 3-space as they do in the plane of World Wallpaper. Further, to remedy the problems with World Wallpaper, we will adopt a different approach than that of Pete. Vectors will guide much of that investigation.
Regard a curving trip along a World Wallpaper path as a function of time. That is, the parameter is time; you imagine walking along the curve with an arm pointing in direction of the walk! At a given time t_0 you are at a certain position on the path. At another time t_1 you are at another position. (These could be the same position, say if you just stood in place for a long time. To simplify a little, pretend you keep moving forward along the path.)
Take the data for the trip as giving a point (v_x(t),v_y(t)) in the plane. This point is in the direction parallel to the direction of the curve when you are at time t. The length \sqrt{v_x(t)^2+v_y(t)^2} is the velocity with of motion at time t. Sometimes science courses, like Physics, ask how you can solve for the position of motion. They are asking you to turn the data here into data for the position (x(t), y(t)) at the time t.
Question: If the trip starts at noon today, and ends
at 3:30, how can you compute the total length of travel in
following the curve where v_x(t)=cos(t) and v_y(t)=sin(t)?
Similarly, how can you do this computation when v_x(t)=t
and v_y(t)=\sqrt{t}?