5.1 Vector Addition and Subtraction: Graphical Methods

Section Key Terms

The Graphical Method of Vector Addition and Subtraction

Recall that a vector is a quantity that has magnitude and direction. For example, displacement, velocity, acceleration, and force are all vectors. In one-dimensional or straight-line motion, the direction of a vector can be given simply by a plus or minus sign. Motion that is forward, to the right, or upward is usually considered to be positive (+); and motion that is backward, to the left, or downward is usually considered to be negative (−).

In two dimensions, a vector describes motion in two perpendicular directions, such as vertical and horizontal. For vertical and horizontal motion, each vector is made up of vertical and horizontal components. In a one-dimensional problem, one of the components simply has a value of zero. For two-dimensional vectors, we work with vectors by using a frame of reference such as a coordinate system. Just as with one-dimensional vectors, we graphically represent vectors with an arrow having a length proportional to the vector’s magnitude and pointing in the direction that the vector points.

Figure 5.2 shows a graphical representation of a vector; the total displacement for a person walking in a city. The person first walks nine blocks east and then five blocks north. Her total displacement does not match her path to her final destination. The displacement simply connects her starting point with her ending point using a straight line, which is the shortest distance. We use the notation that a boldface symbol, such as D, stands for a vector. Its magnitude is represented by the symbol in italics, D, and its direction is given by an angle represented by the symbol $\theta .$ Note that her displacement would be the same if she had begun by first walking five blocks north and then walking nine blocks east.

Figure 5.2 A person walks nine blocks east and five blocks north. The displacement is 10.3 blocks at an angle ${29.1}^{\circ }$ north of east.

The head-to-tail method is a graphical way to add vectors. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the pointed end of the arrow. The following steps describe how to use the head-to-tail method for graphical vector addition.

1. Let the x-axis represent the east-west direction. Using a ruler and protractor, draw an arrow to represent the first vector (nine blocks to the east), as shown in Figure 5.3(a).

   Figure 5.3 The diagram shows a vector with a magnitude of nine units and a direction of 0°.
2. Let the y-axis represent the north-south direction. Draw an arrow to represent the second vector (five blocks to the north). Place the tail of the second vector at the head of the first vector, as shown in Figure 5.4(b).

   Figure 5.4 A vertical vector is added.
3. If there are more than two vectors, continue to add the vectors head-to-tail as described in step 2. In this example, we have only two vectors, so we have finished placing arrows tip to tail.
4. Draw an arrow from the tail of the first vector to the head of the last vector, as shown in Figure 5.5(c). This is the resultant, or the sum, of the vectors.

   Figure 5.5 The diagram shows the resultant vector, a ruler, and protractor.
5. To find the magnitude of the resultant, measure its length with a ruler. When we deal with vectors analytically in the next section, the magnitude will be calculated by using the Pythagorean theorem.
6. To find the direction of the resultant, use a protractor to measure the angle it makes with the reference direction (in this case, the x-axis). When we deal with vectors analytically in the next section, the direction will be calculated by using trigonometry to find the angle.

Vector subtraction is done in the same way as vector addition with one small change. We add the first vector to the negative of the vector that needs to be subtracted. A negative vector has the same magnitude as the original vector, but points in the opposite direction (as shown in Figure 5.6). Subtracting the vector B from the vector A, which is written as A − B, is the same as A + (−B). Since it does not matter in what order vectors are added, A − B is also equal to (−B) + A. This is true for scalars as well as vectors. For example, 5 – 2 = 5 + (−2) = (−2) + 5.

Figure 5.6 The diagram shows a vector, B, and the negative of this vector, –B.

Global angles are calculated in the counterclockwise direction. The clockwise direction is considered negative. For example, an angle of ${30}^{\circ }$ south of west is the same as the global angle ${210}^{\circ },$ which can also be expressed as ${\mathrm{-150}}^{\circ }$ from the positive x-axis.

Using the Graphical Method of Vector Addition and Subtraction to Solve Physics Problems

Now that we have the skills to work with vectors in two dimensions, we can apply vector addition to graphically determine the resultant vector, which represents the total force. Consider an example of force involving two ice skaters pushing a third as seen in Figure 5.7.

Figure 5.7 Part (a) shows an overhead view of two ice skaters pushing on a third. Forces are vectors and add like vectors, so the total force on the third skater is in the direction shown. In part (b), we see a free-body diagram representing the forces acting on the third skater.

In problems where variables such as force are already known, the forces can be represented by making the length of the vectors proportional to the magnitudes of the forces. For this, you need to create a scale. For example, each centimeter of vector length could represent 50 N worth of force. Once you have the initial vectors drawn to scale, you can then use the head-to-tail method to draw the resultant vector. The length of the resultant can then be measured and converted back to the original units using the scale you created.

You can tell by looking at the vectors in the free-body diagram in Figure 5.7 that the two skaters are pushing on the third skater with equal-magnitude forces, since the length of their force vectors are the same. Note, however, that the forces are not equal because they act in different directions. If, for example, each force had a magnitude of 400 N, then we would find the magnitude of the total external force acting on the third skater by finding the magnitude of the resultant vector. Since the forces act at a right angle to one another, we can use the Pythagorean theorem. For a triangle with sides a, b, and c, the Pythagorean theorem tells us that

Applying this theorem to the triangle made by F₁, F₂, and F_tot in Figure 5.7, we get

or

Note that, if the vectors were not at a right angle to each other $({90}^{\circ }$ to one another), we would not be able to use the Pythagorean theorem to find the magnitude of the resultant vector. Another scenario where adding two-dimensional vectors is necessary is for velocity, where the direction may not be purely east-west or north-south, but some combination of these two directions. In the next section, we cover how to solve this type of problem analytically. For now let’s consider the problem graphically.

Worked Example

Adding Vectors Graphically by 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 m in a direction ${49}^{\circ }$ north of east. Then, she walks 23 m heading ${15}^{\circ }$ north of east. Finally, she turns and walks 32 m in a direction ${68}^{\circ }$ south of east.

Strategy

Graphically represent each displacement vector with an arrow, labeling the first A, the second B, and the third C. Make the lengths proportional to the distance of the given displacement and orient the arrows as specified relative to an east-west line. Use the head-to-tail method outlined above to determine the magnitude and direction of the resultant displacement, which we’ll call R.

Solution

(1) Draw the three displacement vectors, creating a convenient scale (such as 1 cm of vector length on paper equals 1 m in the problem), as shown in Figure 5.8.

Figure 5.8 The three displacement vectors are drawn first.

(2) Place the vectors head to tail, making sure not to change their magnitude or direction, as shown in Figure 5.9.

Figure 5.9 Next, the vectors are placed head to tail.

(3) Draw the resultant vector R from the tail of the first vector to the head of the last vector, as shown in Figure 5.10.

Figure 5.10 The resultant vector is drawn .

(4) Use a ruler to measure the magnitude of R, remembering to convert back to the units of meters using the scale. Use a protractor to measure the direction of 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 R is south of the eastward pointing axis (the x-axis), we flip the protractor upside down and measure the angle between the eastward axis and the vector, as illustrated in Figure 5.11.

Figure 5.11 A ruler is used to measure the magnitude of R, and a protractor is used to measure the direction of R.

In this case, the total displacement R has a magnitude of 50 m and points $7^{\circ }$ south of east. Using its magnitude and direction, this vector can be expressed as

$$\text{R = }50\text{ m}$$

5.1

and

$$\theta =7^{\circ }\text{south of east.}$$

5.2

Discussion

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that it does not matter in what order the vectors are added. Changing the order does not change the resultant. For example, we could add the vectors as shown in Figure 5.12, and we would still get the same solution.

Figure 5.12 Vectors can be added in any order to get the same result.

Worked 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 ${66.0}^{\circ }$ north of east from her current location, and then travel 30.0 m in a direction ${112}^{\circ }$ north of east (or ${22.0}^{\circ }$ 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? The two legs of the woman’s trip are illustrated in Figure 5.13.

Figure 5.13 In the diagram, the first leg of the trip is represented by vector A and the second leg is represented by vector B.

Strategy

We can represent the first leg of the trip with a vector A, and the second leg of the trip that she was supposed to take with a vector B. Since the woman mistakenly travels in the opposite direction for the second leg of the journey, the vector for second leg of the trip she actually takes is −B. Therefore, she will end up at a location A + (−B), or A − B. Note that −B has the same magnitude as B (30.0 m), but is in the opposite direction, ${68}^{\circ }({180}^{\circ }-{112}^{\circ })$ south of east, as illustrated in Figure 5.14.

Figure 5.14 Vector –B represents traveling in the opposite direction of vector B.

We use graphical vector addition to find where the woman arrives A + (−B).

Solution

(1) To determine the location at which the woman arrives by accident, draw vectors A and −B.

(2) Place the vectors head to tail.

(3) Draw the resultant vector R.

(4) Use a ruler and protractor to measure the magnitude and direction of R.

These steps are demonstrated in Figure 5.15.

Figure 5.15 The vectors are placed head to tail.

In this case

$$R=23.0\text{m}$$

5.3

and

$$\theta =7.5°\text{south of east.}$$

5.4

Discussion

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

Worked Example

Adding Velocities: A Boat on a River

A boat attempts to travel straight across a river at a speed of 3.8 m/s. The river current flows at a speed v_river of 6.1 m/s to the right. What is the total velocity and direction of the boat? You can represent each meter per second of velocity as one centimeter of vector length in your drawing.

Strategy

We start by choosing a coordinate system with its x-axis parallel to the velocity of the river. Because the boat is directed straight toward the other shore, its velocity is perpendicular to the velocity of the river. We draw the two vectors, v_boat and v_river, as shown in Figure 5.16.

Using the head-to-tail method, we draw the resulting total velocity vector from the tail of v_boat to the head of v_river.

Figure 5.16 A boat attempts to travel across a river. What is the total velocity and direction of the boat?

Solution

By using a ruler, we find that the length of the resultant vector is 7.2 cm, which means that the magnitude of the total velocity is

$${\text{v}}_{\text{tot}}=7.2\text{m/s}.$$

5.5

By using a protractor to measure the angle, we find $\theta ={32.0}^{\circ }\text{.}$

Discussion

If the velocity of the boat and river were equal, then the direction of the total velocity would have been 45°. However, since the velocity of the river is greater than that of the boat, the direction is less than 45° with respect to the shore, or x axis.