Use of Scaled Vector Diagrams to Determine a Resultant

Vector Addition

A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). Observe the following summations of two force vectors:

Use of Scaled Vector Diagrams to Determine a Resultant

The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the head-to-tail method is employed to determine the vector sum or resultant.

Suppose that you were given a map of your local area and a set of 18 directions to follow. Starting at home base, these 18 displacement vectors could be added together in consecutive fashion to determine the result of adding the set of 18 directions. Perhaps the first vector is measured 5 cm, East. Where this measurement ended, the next measurement would begin. The process would be repeated for all 18 directions. Each time one measurement ended, the next measurement would begin. In essence, you would be using the head-to-tail method of vector addition.

The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position. Where the head of this first vector ends, the tail of the second vector begins (thus, head-to-tail method). The process is repeated for all vectors that are being added. Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to real units using the given scale. The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East.

A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below.

1.Print the directions from PHS to your house (or other landmark) using Google, Mapquest, or other internet map/directions site. Look at the map to give an idea of the scale that you will be working with and compare it to your graph paper.

2.Choose a scale and indicate it on the sheet of graph paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper.

3.Pick an appropriate starting location on the graph paper, label it PHS, and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m).

4.Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram.

5.Repeat steps 2 and 3 for all vectors that are to be added

6.Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as “Actual Displacement”.

7.Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m).

8.Measure the direction of the resultant using a protractor and the counterclockwise convention.

9.Using the distance given by the printout (technically cheating- we would really want to calculate this by adding all of the different vector magnitudes) and the time given, calculate the average speed of the trip in miles per hour. Make sure to convert minutes into hours (60 min = 1 hr).

10.Using your calculation of displacement, calculate your average velocity for the trip.