## Coordinate Systems and Reference Frames

A point can be located on an x-, y- coordinate system by its coordinates x and y.

The x-coordinate describes how far along the x-axis the point is located while the y-coordinate describes how far along the y-axis the point is located.

The x- and y-coordinates may be either positive or negative as seen in the examples above.

We will use bold face type to indicate a vector, such as

r. In writing a vector by hand, we will indicate that something is a vector by drawing an arrow above it as . Some such distinguising notation isimportant.Do not write a vector without some distinguishing characteristic or notation.We can describe the point (x,y) as a point located by a

vectorrthat has components of x and y.We can call this x-component, a vector along the x-direction of length x, and indicate that it is a vector by

xor . Likewise, we can call this y-component, a vector along the y-direction of length y, and indicate that it is a vector byyor . Then we can write

r = x + yor

= +Please note that it is _

not_ true to write r = x + y.

r is the length of the vector or the magnitude of the vector. The angle (theta) gives the direction of the vector. How are r, x, y, and related?

Notice that r, x, and y form the sides of a

right triangle. Right triangles are special because of their relation with the trigonometry functions.If we know r, x, and y we can find the trig functions for the angle . More likely, tho', is the situation where we know x and y and want to find r and . Or we may know r and and want to find x and y. All those situations can be handled with the trig functions.

It is common practice to measure the angle from the positive x-axis and to measure it

positivefor acounter-clockwisedirection. The example shown below might be for an angle of = 53^{o}. Then, if r = 10, the components will bex = r cos = (10) (0.6) = 6

y = r sin = (10) (0.8) = 8

Please do

notmemorize these equations. Remember the more basic definitions of the trig functions,sin = opp / hyp

cos = adj / hyp

tan = opp / adj

Then, for this particular case of measuring angle from the x-axis, we haver = hyp

x = adj

y = opp

and

thatmeans thatsin = y / r

cos = x / r

tan = y / x

or

x = r cos

y = r sin

Of course, angle does not need to be limited to the first quadrant. Below might be a diagram for = 150

^{o}. Again, let r = 10 for this numerical example. For that case,x = r cos = (10) ( cos 150

^{o}) = (10) ( - 0.866) = - 8.66y = r sin = (10) ( sin 150

^{o}) = (10) ( 0.500) = 5.00Notice the signs and compare them with the diagram. x = - 8.66 is located to the

leftand y = + 5.00 is locatedup.Always make a diagram!And then compare your results - your answers - with your diagram. Signs are vital and it is all too easy to drop them and simply use the magnitudes.The diagram below might be for r = 10 and = 210

^{o}. For those values, we can find the components byx = r cos = (10) ( cos 210

^{o}) = (10) ( - 0.866) = - 8.66y = r sin = (10) ( sin 210

^{o}) = (10) ( - 0.500) = - 5.00Again, notice the signs and compare them with the diagram. x = - 8.66 is located to the

leftand y = - 5.00 is locateddown.Always make a diagram!And then compare your results - your answers - with your diagram. Signs are vital and it is all too easy to drop them and simply use the magnitudes.We can describe this vector as r = 10, = 210

^{o}as we have above. Or, we can measure angleclockwiseas we have below and describe this vector as r = 10, = - 150^{o}. Either description is as good as the other. They are two ways of describing the same vector or the same point.While it is

commonto measure angles from the x-axis and to measure them aspositiveif they arecounter-clockwise, it is _not_ necessary to do so. Airplane pilots commonly measure angles or directions fromNorth(y) and measure them aspositiveforclockwiseangles. Below might be a location or a vector of r = 10 km, = 53^{o}. In this case, we havex = r sin = (10 km) (sin 53

^{o}) = (10 km) (0.8) = 8 kmy = r cos = (10 km) (cos 53

^{o}) = (10 km) (0.6) = 6 kmNotice that for this arrangement, x is now the

oppositeside (the side of the right triangle opposite the angle ) and y is now theadjacentside (the side of the right triangle adjacent to the angle ). If youalwaysstart with the basic definitions of sine and cosine, you will not have a problem.Giving a location or a vector in terms of the coordinates (x, y) means we are using a

cartesiancoordinate system (or reference frame).Giving a location or a vector in terms of the coordinates (r, ) means we are using a

polarcoordinate system (or reference frame).

(c) Doug Davis, 2001; all rights reserved