Adding Vectors Graphically Using the Head-to-Tail Method: A Woman Takes a Walk

Represent each displacement vector graphically with an arrow, labeling the first $\text{A}$, the second $\text{B}$, and the third $\text{C}$, 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 $\mathbf{\text{R}}$.

(1) Draw the three displacement vectors.

Figure 3.15

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

Figure 3.16

(3) Draw the resultant vector, $\text{R}$.

Figure 3.17

(4) Use a ruler to measure the magnitude of $\mathbf{\text{R}}$, and a protractor to measure the direction of $\text{R}$. 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.

Figure 3.18

In this case, the total displacement $\mathbf{\text{R}}$ is seen to have a magnitude of 50.0 m and to lie in a direction $\mathrm{7.0º}$ south of east. By using its magnitude and direction, this vector can be expressed as $R=\text{50.0 m}$ and $\theta =7\text{.}\text{0º}$ south of east.

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 3.19 and we will still get the same solution.

Figure 3.19

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.

(This is true for the addition of ordinary numbers as well—you get the same result whether you add $\mathbf{\text{2}}+\mathbf{\text{3}}$ or $\mathbf{\text{3}}+\mathbf{\text{2}}$, for example).