2.1 Scalars and vectors  (Page 7/29)

If we need to add three or more vectors, we repeat the parallelogram rule for the pairs of vectors until we find the resultant of all of the resultants. For three vectors, for example, we first find the resultant of vector 1 and vector 2, and then we find the resultant of this resultant and vector 3. The order in which we select the pairs of vectors does not matter because the operation of vector addition is commutative and associative (see [link] and [link] ). Before we state a general rule that follows from repetitive applications of the parallelogram rule, let’s look at the following example.

Suppose you plan a vacation trip in Florida. Departing from Tallahassee, the state capital, you plan to visit your uncle Joe in Jacksonville, see your cousin Vinny in Daytona Beach, stop for a little fun in Orlando, see a circus performance in Tampa, and visit the University of Florida in Gainesville. Your route may be represented by five displacement vectors $\overrightarrow{A},$ $\overrightarrow{B}$ , $\overrightarrow{C}$ , $\overrightarrow{D}$ , and $\overrightarrow{E}$ , which are indicated by the red vectors in [link] . What is your total displacement when you reach Gainesville? The total displacement is the vector sum of all five displacement vectors, which may be found by using the parallelogram rule four times. Alternatively, recall that the displacement vector has its beginning at the initial position (Tallahassee) and its end at the final position (Gainesville), so the total displacement vector can be drawn directly as an arrow connecting Tallahassee with Gainesville (see the green vector in [link] ). When we use the parallelogram rule four times, the resultant $\overrightarrow{R}$ we obtain is exactly this green vector connecting Tallahassee with Gainesville: $\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}+\overrightarrow{D}+\overrightarrow{E}$ .

Drawing the resultant vector of many vectors can be generalized by using the following tail-to-head geometric construction    . Suppose we want to draw the resultant vector $\overrightarrow{R}$ of four vectors $\overrightarrow{A}$ , $\overrightarrow{B}$ , $\overrightarrow{C}$ , and $\overrightarrow{D}$ ( [link] (a)). We select any one of the vectors as the first vector and make a parallel translation of a second vector to a position where the origin (“tail”) of the second vector coincides with the end (“head”) of the first vector. Then, we select a third vector and make a parallel translation of the third vector to a position where the origin of the third vector coincides with the end of the second vector. We repeat this procedure until all the vectors are in a head-to-tail arrangement like the one shown in [link] . We draw the resultant vector $\overrightarrow{R}$ by connecting the origin (“tail”) of the first vector with the end (“head”) of the last vector. The end of the resultant vector is at the end of the last vector. Because the addition of vectors is associative and commutative, we obtain the same resultant vector regardless of which vector we choose to be first, second, third, or fourth in this construction.