### Graphical Method Solution

It turns out there are two approaches to examine and solve vector problems. The first is known as the graphical method. This type of method is accomplished by using a ruler and protractor and some kind of directional system to determine the magnitude and direction of vectors, respectively. A problem that occurs when using this method is that many times the vectors stated in the problem are of very large magnitude. For example, a problem may ask to find the resultant vector of two vectors whose magnitudes are 55 meters at 35º and 215º, respectively. How does one draw a 55 meter vector on an 8" X 11" sheet of paper? It is done by "scaling" the vector(s) to an appropriate size so that they are able to be analyzed. By "scaling" it is meant that a vector that is 55 meters can be reduced proportionally to a size that is more appropriate. For example a vector with magnitude of 55 meters can be scaled to 5.5 centimeters on a sheet of paper using a scale of 10 meters = 1.0 centimeters. Vectors that are scaled can now be added (or subtracted) using the familiar head-to tail addition that was learned previously.

How does one add vectors on a sheet of paper? First you must get a ruler and protractor. Scale the vectors so that they will fit on a sheet of paper. Now you must set a directional system. Why do you need a directional system to find our vectors directions? Think about it this way: How do you know where 180º is on your sheet of paper? Neither you nor I know, at the moment. But we do know where our Cardinal Directions are. By Cardinal directions we are referring to North/South/East/West. The easiest way to set a directional system is to locate the direction North first and hence the three other directions as well. Then place on your paper a symbol similar to the one given below:

We also know that by standard we refer to the direction North as being at 90º. Thus if North is at 90 degrees then going clockwise East is 0º (or 360 degrees). West is at 180º and South is at 270º. If we start at zero (East) degrees we increase our angle by going counterclockwise. We come to a full cycle when we reach East again. Thus East represents both 0º and 360º. Now that we have set a directional system we can just use a ruler and protractor to set the magnitude and direction. Using head-to-tail addition we can add as many vectors as we wish. After adding all our vectors to find the resultant vector we make a directed line segment starting from the *tail* of the *first* vector to the *head* of the *last* vector. To find the magnitude we just use a ruler to measure and a protractor to find the direction.

It is necessary to note that this method of analyzing vectors is a very *intuitive* way of understanding vectors. However, it does not yield extremely accurate answers. Why? There are many ways to introduce error. For example, measurement error, not rounding properly, and quality of the measuring instruments. (They are only accurate to a certain degree.)

What, then is a more accurate means of analyzing vectors? Well, we know mathematics is a very precise subject. How then could we incorporate the mathematics we learned to analyze vectors? Ordered Pairs.