Resultant Vector Physics

Resultant Vector: When two or more vectors are added they yield the sum or resultant vector. A resultant vector isthe result or sum of vector addition. Vector addition is some what different from addition ofpure numbers unless the addition takes place alonga straight line. In the latter case, it reduces to thenumber line of standards or scale addition. Forexample, if one walks five miles east and thenthree miles east, he is eight miles from his starting point. On a graph (Figure 8), the sum of the two vectors, i.e., the sum of the five miles plus the three mile displacement, is the total or resultant displacement of eight miles.
Similarly, if one walks five miles east and then three miles west, the resultant displacement is two miles east (Figure 9).The vector diagrams of Figure 8 and Figure 9 are basically scale diagrams of what is happening in the real world of addition of vector quantities.
The vector diagrams of Figure 8 andFigure 9 are basically scale diagrams ofwhat is happening in the real world of addition of vector quantities.
Consider next the addition of vector quantities which are not in a straight line. For example, consider the resultant displacement when a person travels four miles east and then three miles north. Again a scale drawing (Figure 10) is in order. Use a scale of 1 inch = 1 mile.
When drawing a scale drawing, one draws a straight line from the origin C to the final position B to represent the net or resultant displacement. Drawing the straight line CB and measuring its length, one should obtain about 5 inches. Then, since the scale of the drawing is 1 inch = 1 mile, this is used as a conversion factor giving 5 inches × (1 miles/inch) = 5 miles as the inch displacement.

Using a protractor or trigonometry, the acute angle ACB can be determined to be about 37° . Thus, the resultant (or vector sum) of traveling 4 miles east plus 3 miles north is a displacement of 5 miles at 37 degrees north of east.
It is left as an exercise for the reader to show that vector addition is commutative, using the above example. Specifically, make a scale drawing showing that traveling 3 miles north and then 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 three miles east and then three miles north and then three miles west and arrive at a point three miles north of the starting point. The sum of these three displacements is a resultant displacement of 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 south and then 12 miles east and then 25 miles north. Make a scale drawing and determine the magnitude and direction of the resultant displacement. A scale of 2 miles per centimeter or 4 miles per inch will fit the drawing on standard paper.

Answer: About 20 miles at 53° north of east.

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