4.1 Displacement and velocity vectors  (Page 3/7)

[link] can also be written in terms of the components of $\overrightarrow{v}(t).$ Since

we can write

where

If only the average velocity is of concern, we have the vector equivalent of the one-dimensional average velocity for two and three dimensions:

The independence of perpendicular motions

When we look at the three-dimensional equations for position and velocity written in unit vector notation, [link] and [link] , we see the components of these equations are separate and unique functions of time that do not depend on one another. Motion along the x direction has no part of its motion along the y and z directions, and similarly for the other two coordinate axes. Thus, the motion of an object in two or three dimensions can be divided into separate, independent motions along the perpendicular axes of the coordinate system in which the motion takes place.

To illustrate this concept with respect to displacement, consider a woman walking from point A to point B in a city with square blocks. The woman taking the path from A to B may walk east for so many blocks and then north (two perpendicular directions) for another set of blocks to arrive at B . How far she walks east is affected only by her motion eastward. Similarly, how far she walks north is affected only by her motion northward.