3.2 Vector Addition and Subtraction: Graphical Methods

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

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 $\text{49.0º}$ north of east. Then, she walks 23.0 m heading $\text{15.0º}$ 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 $\text{A}$, 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 $\mathbf{\text{R}}$.

Solution

(1) Draw the three displacement vectors.

Figure 3.14

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

Figure 3.15

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

Figure 3.16

(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 3.17

In this case, the total displacement $\mathbf{\text{R}}$ is seen to have a magnitude of 50.8 m and to lie in a direction $\mathrm{5.47º}$ south of east. By using its magnitude and direction, this vector can be expressed as $R=\text{50.8 m}$ and $\theta =5\text{.}\text{47º}$ 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 3.18 and we will still get the same solution.

Figure 3.18

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).

Subtracting Vectors Graphically: A Woman Sailing a Boat

A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction $\text{66.0º}$ north of east from her current location, and then travel 30.0 m in a direction $\text{112º}$ north of east (or $\text{22.0º}$ west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.

Figure 3.20

Strategy

We can represent the first leg of the trip with a vector $\mathbf{\text{A}}$, and the second leg of the trip with a vector $\mathbf{\text{B}}$. The dock is located at a location $\mathbf{\text{A}}+\mathbf{\text{B}}$. If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance $B$ (30.0 m) in the direction $\mathrm{180º}–\mathrm{112º}=\mathrm{68º}$ south of east. We represent this as $\mathbf{\text{–B}}$, as shown below. The vector $\mathbf{\text{–B}}$ has the same magnitude as $\mathbf{\text{B}}$ but is in the opposite direction. Thus, she will end up at a location $\mathbf{\text{A}}+(\mathbf{\text{–B}})$, or $\mathbf{\text{A}}–\mathbf{\text{B}}$.

Figure 3.21

We will perform vector addition to compare the location of the dock, $\text{A }\text{+ }\mathbf{B}$, with the location at which the woman mistakenly arrives, $\text{A + }(\text{–B})$.

Solution

(1) To determine the location at which the woman arrives by accident, draw vectors $\mathbf{\text{A}}$ and $\mathbf{\text{–B}}$.

(2) Place the vectors head to tail.

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

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

Figure 3.22

In this case, $R=\text{23}\text{.}\text{0 m}$ and $\theta =7\text{.}\text{5º}$ south of east.

(5) To determine the location of the dock, we repeat this method to add vectors $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$. We obtain the resultant vector $\mathbf{\text{R}}\text{'}$:

Figure 3.23

In this case $R\text{ = 52.9 m}$ and $\theta =\text{90.1º}$  north of east.

We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.

Discussion

Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.