M1.2 Displacement and velocity – advanced level
To define displacement more rigorously we must introduce a coordinate system. The one that is most popular in Physics and easy to use is the Cartesian coordinate system.
Fig M1.2 Right-handed Cartesian coordinate system.
On Fig. M1.2 there is a three dimensional coordinate system used in Physics. It consists of three mutually perpendicular axes. With the orientation of axes given in this figure it is a so called right-handed coordinate system. The other type may be a left-handed coordinate system. The distinction arises from the fact that once two perpendicular directions have been chosen for the x and y axes, there are two possible choices for the positive direction of z axis.
What does “right-handed” mean? Suppose you have an ordinary screw lying along the z axis and it is mounted in the nut. Now turn it as if by the screwdriver, to rotate it in the direction from x to y. The direction in which it moves, is the direction of the positive z axis in a right-handed coordinate system. This applies for a screw that is tightened by turning it clockwise, with a so-called right-hand thread. This is the most common kind of screw.
Alternatively, imagine you are gripping the z axis in a fist in such a way that your firngers curl round from x to y axis. The direction your thumb points in is defined as the direction of the positive z axis. If you use the fist of your right hand, you have a right-hand coordinate system. The same procedure repeated with the left hand fist will define a left-hand coordinate system.
The position of any point A in space, relative to the origin of the Cartesian coordinate system is given by a set of three numbers xA, yA, and zA Fig. M1.3.
Fig. M1.3 Position of point A given by vector 
.
For denoting the position in a coordinate system a widely used notation is (x,
y, z) – three symbols or numbers in parenthesis. If we draw an arrow from
the origin of coordinate system to point A this will be the vector defining the
position of point A. It is usually denoted 
. This
vector notation is very handy, but any algebraic operations must be performed
on vector components. When using vector notation it is customary to denote
directions of coordinate axis as unit vectors 
. With this
convention the vector from Fig. M1.3
can be written in the form
Position of any
other point B will be defined by analogically constructed vector 
.
For the movement of object from point A to point B, the displacement of this
object will be defined as
To calculate this displacement we must express each vector through its components, then perform algebra on these components and build the resultant vector from the new components. In this case the calculations may be written as:
(M1.7)With the notation introduced above, the velocity of object moving from A to B is
. (M1.8)This is the average velocity for the trajectory being a straight line. We will now introduce the definition of instantaneous velocity as
This is a
velocity measured for a moving point, when time allowed for displacement approaches
zero. The Equation M1.9 is a definition of the derivative of displacement with
respect to time and is denoted 
( read it
d, r over d, t). So the final definition of velocity may be written in the form
(M1.10)This definition of velocity is valid for any type of motion, not only for motion along a straight line. It is easy to show that Formula M1.3 is a special case of Formula M1.10. We will return to this general definition of velocity in paragraphs concerning motion in two and three dimensions.