VECTORS: RESULTANTS AND COMPONENTSVectorsCP-02Page 10Rev. 0It is left as an exercise for the student to show that vector addition is commutative, using theabove example. Specifically, make a scale drawing showing that traveling 3 miles north andthen 4 miles east yields the same resultant as above.It is also reasonably obvious that more than two vectors can be added. One can travel threemiles east and then three miles north and then three miles west and arrive at a point three milesnorth of the starting point. The sum of these three displacements is a resultant displacementof three miles north. (If this is not immediately apparent, sketch it.)A student problem is to find the net or resultant displacement if a person travels 9 miles southand then 12 miles east and then 25 miles north. Make a scale drawing and determine themagnitude and direction of the resultant displacement. A scale of 2 miles per centimeter or4 miles per inch will fit the drawing on standard paper.Answer:About 20 miles at 53 north of east.oVectorComponentsComponentsof a vector are vectors, which when added, yield the vector. For example, asshown in the previous section (Figure 10), traveling 3 miles north and then 4 miles east yieldsa resultant displacement of 5 miles, 37 north of east. This example demonstrates thatocomponent vectors of any two non-parallel directions can be obtained for any resultant vectorin the same plane. For the purposes of this manual, we restrict our discussions to twodimensional space. The student should realize that vectors can and do exist in three dimensionalspace.One could write an alternate problem: "If I am 5 miles from where I started northeast along aline 37 N of east, how far north and how far east am I from my original position?" Drawing thisoon a scale drawing, the vector components in the east and north directions can be measured tobe about 4 miles east and 3 miles north. These two vectors are the components of the resultantvector of 5 miles, 37 north of east.oComponent vectors can be determined by plotting them on a rectangular coordinate system. Forexample, a resultant vector of 5 units at 53 can be broken down into its respective x and yomagnitudes. The x value of 3 and the y value of 4 can be determined using trigonometry orgraphically. Their magnitudes and position can be expressed by one of several conventionsincluding: (3,4), (x=3, y=4), (3 at 0 , 4 at 90 ), and (5 at 53 ). In the first expression, the firsto o oterm is the x-component (F ), and the second term is the y-component (F ) of the associatedx yresultant vector.