Vector Addition and Subtraction: Graphical Methods, Motion in Two Dimensions | Physics

Learning objectives

Section Key Terms

graphical method	head (of a vector)	head-to-tail method	resultant
resultant vector	tail	vector addition	vector subtraction

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_05_01_city 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 $θ .$ Note that her displacement would be the same if she had begun by first walking five blocks north and then walking nine blocks east.

View original figure (external; image not reproduced in this text edition)

Original figure description: A map is shown over top x and y-axes. The origin is labeled Starting Point. The x-axis represents east and has a scale from zero to nine in increments of one. The y-axis represents north and has a scale from zero to five in increments of one. Lines show that a person walks nine blocks east and five blocks north. A displacement vector is plotted from the origin to the destination of nine, five on the axes.

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.

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_05_01_vectora (a).

View original figure (external; image not reproduced in this text edition)

Original figure description: The diagram shows a vector along an x-axis with a magnitude of nine units and a direction of 0°. An unlabeled y-axis is also shown.

The diagram shows a vector with a magnitude of nine units and a direction of 0°.
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_05_01_vectorb (b).

View original figure (external; image not reproduced in this text edition)

Original figure description: The diagram shows a vector along an x-axis with a magnitude of nine units and a direction of 0°. A vertical vector is also shown at nine units with a height of five units. A y-axis is also shown with ten tick marks.

A vertical vector is added.
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.
Draw an arrow from the tail of the first vector to the head of the last vector, as shown in Figure_05_01_vectorc (c). This is the resultant, or the sum, of the vectors.

View original figure (external; image not reproduced in this text edition)

Original figure description: The diagram shows a vector along an x-axis with a magnitude of nine units and a direction of 0°. A vertical vector is also shown at nine units with a height of five units. A y-axis is shown with ten tick marks. A resultant vector is drawn, forming a right triangle with the other two vectors. This vector is labeled D, has a magnitude of ten point three units, and forms a twenty-nine point one degree angle with the x-axis.

The diagram shows the resultant vector, a ruler, and protractor.
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.
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_05_01_negative). 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.

View original figure (external; image not reproduced in this text edition)

Original figure description: Two vectors are shown. Vector B points upward at an angle. Vector negative B is parallel to vector B and points downward.

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 ${−150}^{\circ}$ 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_05_01_skaters.

View original figure (external; image not reproduced in this text edition)

Original figure description: Two diagrams are shown. Diagram a is an overhead view of two skaters, one on the left and one below, pushing on a third skater in the center. A force vector is shown on each of the pushing skaters pointing from the skater's shoulder to the skater in the center. A right triangle is drawn using the force vectors, with F tot as the hypotenuse, F one as the bottom leg, and F two as the right leg. Diagram b is a free body diagram with F one pointing to the right and F two pointing up. Both vectors have the same magnitude.

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_05_01_skaters 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

\begin{matrix} a^{2} + b^{2} = c^{2} \\ c = \sqrt{a^{2} + b^{2}} . \end{matrix}

Applying this theorem to the triangle made by F₁, F₂, and F_tot in Figure_05_01_skaters, we get

F_{tot}^{2} = \sqrt{F_{1}^{2} + F_{1}^{2}},

F_{tot} = \sqrt{{(400 N)}^{2} + {(400 N)}^{2}} = 566 N.

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.

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.

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_05_01_displace.

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Original figure description: Three displacement vectors are drawn on separate x-y axes. Vector A points upward from the origin and is twenty-five meters with an angle of forty-nine degrees. Vector B points upward from the origin and is twenty-three meters with an angle of fifteen degrees. Vector C points downward from the origin and is thirty-two meters with an angle of sixty-eight degrees.

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_05_01_tall.

View original figure (external; image not reproduced in this text edition)

Original figure description: Three vectors, A, B, and C are shown. They form three sides of a polygon, with an x-axis forming the fourth side. The arrow of vector A touches the end of vector B and has an x-y-axis shown at the touching point. The arrow of vector B touches the end of vector C and has an x-y-axis shown at the touching point.

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_05_01_result.

View original figure (external; image not reproduced in this text edition)

Original figure description: Vectors A, B, C, and R form a four-sided figure, with vector R intersecting an x-axis. The magnitude of vector R is fifty meters and vector R forms a seven degree angle with the x-axis.

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_05_01_measure.

View original figure (external; image not reproduced in this text edition)

Original figure description: Vectors A, B, C, and R form a four-sided figure, with vector R intersecting an x-axis. A ruler and a protractor are pictured below the figure. The magnitude of vector R is fifty meters and the direction of vector R is ninety degrees.

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

$R = 50 m$

and

$θ = 7^{\circ} south of east.$

Discussion

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that 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_05_01_alt, and we would still get the same solution.

View original figure (external; image not reproduced in this text edition)

Original figure description: Vectors A, B, C, and R are shown. Angle CR is against the origin of x and y-axes and vector R has a length of 50 centimeters. Vector R forms a seven degree angle with the x-axis. Vectors C and A form an angle, and vectors B and R form an angle.

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

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_05_01_magnitude.

View original figure (external; image not reproduced in this text edition)

Original figure description: An x-y axis is shown for reference. Vector A points upward from the x-axis is twenty-seven point five meters, and has an angle of sixty-six degrees. Vector B points upward from the x-axis is thirty meters, and has an angle of one hundred twelve degrees.

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_05_01_inclined.

View original figure (external; image not reproduced in this text edition)

Original figure description: An x-y-axis is shown on the left for reference. On the right, vector negative B points downward and intersects the x-axis, forming an angle that measures sixty-eight degrees and a supplementary angle that measures one hundred twelve degrees.

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_05_01_connected.

View original figure (external; image not reproduced in this text edition)

Original figure description: An x-y axis is shown on the left. On the right, vectors A, negative B, and R form a triangle. R and the x-axis form an angle with a measure of seven point five degrees. A minus B equals R and R equals twenty-three meters.

The vectors are placed head to tail.

In this case

$R = 23.0 m$

and

$θ = 7.5 ° south of east.$

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.

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_05_01_river.

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.

View original figure (external; image not reproduced in this text edition)

Original figure description: A river is shown with a boat traveling across it. Three images of the boat are shown to indicate the boat's motion in diagonal path across the river from the bottom left to the upper right corner. Three vectors are drawn to form a right triangle. The hypotenuse intersects the path of the boat and is labeled V tot. The left leg of the triangle is labeled V boat, three point eight meters per second. The upper leg of the triangle is labeled V river at one point six one meters per second. An x-y axis is in the bottom left corner for reference.

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

$v_{tot} = 7.2 m/s .$

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

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.

Practice Problems

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Vector Addition

In this simulation, you will experiment with adding vectors graphically. Click and drag the red vectors from the Grab One basket onto the graph in the middle of the screen. These red vectors can be rotated, stretched, or repositioned by clicking and dragging with your mouse. Check the Show Sum box to display the resultant vector (in green), which is the sum of all of the red vectors placed on the graph. To remove a red vector, drag it to the trash or click the Clear All button if you wish to start over. Notice that, if you click on any of the vectors, the $| R |$ is its magnitude, $θ$ is its direction with respect to the positive x-axis, R_x is its horizontal component, and R_y is its vertical component. You can check the resultant by lining up the vectors so that the head of the first vector touches the tail of the second. Continue until all of the vectors are aligned together head-to-tail. You will see that the resultant magnitude and angle is the same as the arrow drawn from the tail of the first vector to the head of the last vector. Rearrange the vectors in any order head-to-tail and compare. The resultant will always be the same.

Original figure description: This simulation provides the tools to experiment with adding vectors graphically.

True or FalseThe more long, red vectors you put on the graph, rotated in any direction, the greater the magnitude of the resultant green vector.

True
False

Section Summary

The graphical method of adding vectors $A$ and $B$ involves drawing vectors on a graph and adding them by using the head-to-tail method. The resultant vector $R$ is defined such that A + B = R. The magnitude and direction of $R$ are then determined with a ruler and protractor.
The graphical method of subtracting vectors A and B involves adding the opposite of vector B, which is defined as −B. In this case, $A - B = A + (- B) = R .$ Next, use the head-to-tail method as for vector addition to obtain the resultant vector $R$ .
Addition of vectors is independent of the order in which they are added; A + B = B + A.
The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
Variables in physics problems, such as force or velocity, can be represented with vectors by making the length of the vector proportional to the magnitude of the force or velocity.
Problems involving displacement, force, or velocity may be solved graphically by measuring the resultant vector’s magnitude with a ruler and measuring the direction with a protractor.

Check Your Understanding

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True or FalseA person walks 2 blocks east and 5 blocks north. Another person walks 5 blocks north and then two blocks east. The displacement of the first person will be more than the displacement of the second person.

True
False

Concept Items

There is a vector $\vec{A}$ , with magnitude 5 units pointing towards west and vector $\vec{B}$ , with magnitude 3 units, pointing towards south. Using vector addition, calculate the magnitude of the resultant vector.

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Problems

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Critical Thinking

True or FalseA person is following a set of directions. He has to walk 2 km east and then 1 km north. He takes a wrong turn and walks in the opposite direction for the second leg of the trip. The magnitude of his total displacement will be the same as it would have been had he followed directions correctly.

True
False

Test Prep Multiple Choice

True or False—We can use Pythagorean theorem to calculate the length of the resultant vector obtained from the addition of two vectors which are at right angles to each other.

True
False

True or FalseThe direction of the resultant vector depends on both the magnitude and direction of added vectors.

True
False

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Test Prep Short Answer

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Consider six vectors of 2 cm each, joined from head to tail making a hexagon. What would be the magnitude of the addition of these vectors?

Zero
Six
Eight
Twelve

Two people pull on ropes tied to a trolley, each applying 44 N of force. The angle the ropes form with each other is 39.5°. What is the magnitude of the net force exerted on the trolley?

0.0 N
79.6 N
82.8 N
88.0 N

Test Prep Extended Response

True or False—For vectors the order of addition is important.

True
False

Consider five vectors a, b, c, d, and e. Is it true or false that their addition always results in a vector with a greater magnitude than if only two of the vectors were added?

True
False

Glossary

graphical method: drawing vectors on a graph to add them using the head-to-tail method
head (of a vector): the end point of a vector; the location of the vector’s arrow; also referred to as the tip
head-to-tail method: a method of adding vectors in which the tail of each vector is placed at the head of the previous vector
resultant: the sum of the a collection of vectors
resultant vector: the vector sum of two or more vectors
tail: the starting point of a vector; the point opposite to the head or tip of the arrow
vector addition: adding together two or more vectors

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