Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction 49.0º49.0º size 12{"49" "." "0º"} {} north of east. Then, she walks 23.0 m heading 15.0º15.0º size 12{"15" "." "º°"} {} north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east.

*Strategy*

Represent each displacement vector graphically with an arrow, labeling the first AA size 12{A} {}, the second BB size 12{B} {}, and the third CC size 12{C} {}, making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted RR size 12{R} {}.

*Solution*

(1) Draw the three displacement vectors.

(2) Place the vectors head to tail retaining both their initial magnitude and direction.

(3) Draw the resultant vector, RR size 12{R} {}.

(4) Use a ruler to measure the magnitude of RR size 12{R} {}, and a protractor to measure the direction of RR size 12{R} {}. While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.

In this case, the total displacement RR size 12{R} {} is seen to have a magnitude of 50.0 m and to lie in a direction 7.0º7.0º size 12{7 "." 0°} {} south of east. By using its magnitude and direction, this vector can be expressed as
R
=
50.0 m
R
=
50.0 m
size 12{R" = 50" "." "0 m"} {}
and θ=7.0ºθ=7.0º size 12{θ=7 "." "0°"} {} south of east.

*Discussion*

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in Figure 12 and we will still get the same solution.

Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative. Vectors can be added in any order.

A+B=B+A.A+B=B+A. size 12{"A+B=B+A"} {}

(1)(This is true for the addition of ordinary numbers as well—you get the same result whether you add 2+32+3 size 12{"2+3"} {}
or
3+23+2 size 12{"3+2"} {}, for example).

Comments:"This introductory, algebra-based, two-semester college physics book is grounded with real-world examples, illustrations, and explanations to help students grasp key, fundamental physics concepts. […]"