As we’ve seen in Ex. 6.3, performing vector arithmetic using the graphical method is straightforward, but a bit tedious and time-consuming since everything must be done to scale using rulers and protractors.  Nevertheless, this method can give us good results for vector arithmetic.  However, it takes no great stretch of the imagination to realize that this method can get very old and very messy if we have many vectors to add or subtract!

The analytical method of performing vector arithmetic is also straightforward, and does not get much messier as more vectors are involved.  This method also has the advantage that the answers are as precise as the information that you are given, provided that you retain the proper number of decimal places in your calculations.  (You should keep as least as many as are in the numbers given.  Your answers should, in general, be kept to only as many significant figures as are in the least significant data that you are working with. You’ll have more practice with this in the Problems Lab than you will in lecture, but it’s something that you should always keep in mind when performing calculations with experimental data.)

The analytical method of performing vector arithmetic goes as follows.  First, find the components of each of the vectors involved in the equation being performed. Next, substitute, for example, the x-component of each of the vectors involved into the equation.  The result will give you the x-component of the resulting vector.  Do the same thing with the y-components,  etc. What this then gives you are the components of the resulting vector.  To find the vector’s magnitude and direction, all you then need do is follow the procedure demonstrated in Ex. 6.2.

For example, if the vector D is given by

then the x- and y-components of the vector D are computed form the components of the vectors A, B, and C as follows:

These components of the vector D can then be used to find its magnitude and direction. This process is demonstrated in the next example, Ex. 6.4.