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Adding Vectors

Vectors, like scalars, can be added together. However, adding vectors and scalars is very different. The direction a vector has affects the final magnitude of the addition of the two vectors. When we add two or more vectors together, their addition results in a final magnitude and direction. This is called the Resultant Vector or Vector Sum. The easiest best way to intuitively understand how to add vectors is to use add them graphically. In other words, using a ruler and protractor. To add two vectors together, place the first vector where you want it and place the second vector's tail on the tip of the head of the first vector. This is called head-to-tail addition. Regardless of how many vectors you have one still uses head-to-tail addition to add vectors graphically. To find the Resultant Vector one just needs to add a line from the starting point of the tail of the very FIRST vector to the head of the very LAST vector. An arrowhead is placed on the ending point of the resultant vector to indicate direction. Thus using a ruler and protractor one can find the resultant magnitude and direction. Here are two examples:

Example I: Adding vectors A and B to get resultant (vector R) Example II: Adding vectors A, B, C, and D to get resultant (vector R)

When we write vectors it is best to use letters to indicate each vector. These letters are distinguished from scalar quantities by adding a small arrow over the top of the letter. Or to indicate a vector quantity on a computer or word processor one puts the letter in boldfaced print. In the above example, head-to-tail addition was used in adding each vector (A, B, C, D). The Resultant Vector is the directed line segment in dotted blue. Notice where it starts and ends. Thus one can see how direction affects the final magnitude of a Resultant Vector. If one were to just add the magnitudes of vector A and vector B in the first example (I) and very different answer would be obtained. Can you see? To write an equation for the adding of two or more vectors we do the following (addition of example I and II are being used):

	I.  R = A + B
	II. R = A + B + C + D