5.1 Vector Addition and Subtraction: Graphical Methods

Section learning objectives.

By the end of this section, you will be able to do the following:

  • Describe the graphical method of vector addition and subtraction
  • Use the graphical method of vector addition and subtraction to solve physics problems

Teacher Support

The learning objectives in this section will help your students master the following standards:

  • (E) develop and interpret free-body force diagrams.

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.

[BL] [OL] Review vectors and free body diagrams. Recall how velocity, displacement and acceleration vectors are represented.

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 θ . 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.

Tips For Success

In this text, we represent a vector with a boldface variable. For example, we represent a force with the vector F , which has both magnitude and direction. The magnitude of the vector is represented by the variable in italics, F , and the direction of the variable is given by the angle θ . θ .

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 .

  • 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.
  • 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.

[AL] Ask two students to demonstrate pushing a table from two different directions. Ask students what they feel the direction of resultant motion will be. How would they represent this graphically? Recall that a vector’s magnitude is represented by the length of the arrow. Demonstrate the head-to-tail method of adding vectors, using the example given in the chapter. Ask students to practice this method of addition using a scale and a protractor.

[BL] [OL] [AL] Ask students if anything changes by moving the vector from one place to another on a graph. How about the order of addition? Would that make a difference? Introduce negative of a vector and vector subtraction.

Watch Physics

Visualizing vector addition examples.

This video shows four graphical representations of vector addition and matches them to the correct vector addition formula.

  • Yes, if we add the same two vectors in a different order it will still give the same resultant vector.
  • No, the resultant vector will change if we add the same vectors in a different order.

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.

Global angles are calculated in the counterclockwise direction. The clockwise direction is considered negative. For example, an angle of 30 ∘ 30 ∘ south of west is the same as the global angle 210 ∘ , 210 ∘ , which can also be expressed as −150 ∘ −150 ∘ 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 .

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 F 1 , F 2 , and F tot in Figure 5.7 , we get

Note that, if the vectors were not at a right angle to each other ( 90 ∘ ( 90 ∘ 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 ∘ 49 ∘ north of east. Then, she walks 23 m heading 15 ∘ 15 ∘ north of east. Finally, she turns and walks 32 m in a direction 68 ∘ 68 ∘ south of east.

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 .

(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 .

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

(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 .

(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 .

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

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.

[BL] [OL] [AL] Ask three students to enact the situation shown in Figure 5.8 . Recall how these forces can be represented in a free-body diagram. Giving values to these vectors, show how these can be added graphically.

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 ∘ 66.0 ∘ north of east from her current location, and then travel 30.0 m in a direction 112 ∘ 112 ∘ north of east (or 22.0 ∘ 22.0 ∘ 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 .

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 ∘ ( 180 ∘ − 112 ∘ ) 68 ∘ ( 180 ∘ − 112 ∘ ) south of east, as illustrated in Figure 5.14 .

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

(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 .

In this case

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.

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 .

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

By using a protractor to measure the angle, we find θ = 32.0 ∘ . θ = 32.0 ∘ .

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.

Teacher Demonstration

Plot the way from the classroom to the cafeteria (or any two places in the school on the same level). Ask students to come up with approximate distances. Ask them to do a vector analysis of the path. What is the total distance travelled? What is the displacement?

Practice Problems

Virtual physics, 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 | | R | is its magnitude, θ 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.

Grasp Check

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

Check Your Understanding

  • backward and to the left
  • backward and to the right
  • forward and to the right
  • forward and to the left

True or False—A 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.

Use the Check Your Understanding questions to assess whether students achieve the learning objectives for this section. If students are struggling with a specific objective, the Check Your Understanding will help identify which objective is causing the problem and direct students to the relevant content.

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This chapter will formally introduce vectors and will describe the various operations associated with them. It will serve a prelude to relative motion and motion in a plane.

  • 4.1: Introduction to Vectors This page introduces vectors for use in physics problems. It then proceeds to derive the parallelogram law of vector addition.
  • 4.2: Unit Vectors and Vector Resolution

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physics vectors homework

These rules for summing vectors were applied to free-body diagrams in order to determine the net force (i.e., the vector sum of all the individual forces). Sample applications are shown in the diagram below.

In this unit, the task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. For example, a vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below.

There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are:

  • the Pythagorean theorem and trigonometric methods
  • the head-to-tail method using a scaled vector diagram

The Pythagorean Theorem

The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

To see how the method works, consider the following problem:

Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

This problem asks to determine the result of adding two displacement vectors that are at right angles to each other. The result (or resultant) of walking 11 km north and 11 km east is a vector directed northeast as shown in the diagram to the right. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant (i.e., the hypotenuse of the right triangle).

The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6 km. Later , the method of determining the direction of the vector will be discussed.

R 2 = (5) 2 + (10) 2

R = SQRT (125)

R = 11.2 km

R 2 = (30) 2 + (40) 2

R = SQRT (2500)

Using Trigonometry to Determine a Vector's Direction

The direction of a resultant vector can often be determined by use of trigonometric functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA from their course in trigonometry. SOH CAH TOA is a mnemonic that helps one remember the meaning of the three common trigonometric functions - sine, cosine, and tangent functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The tangent function relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The three equations below summarize these three functions in equation form.

These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement. The process begins by the selection of one of the two angles (other than the right angle) of the triangle. Once the angle is selected, any of the three functions can be used to find the measure of the angle. Write the function and proceed with the proper algebraic steps to solve for the measure of the angle. The work is shown below.

Once the measure of the angle is determined, the direction of the vector can be found. In this case the vector makes an angle of 45 degrees with due East. Thus, the direction of this vector is written as 45 degrees. (Recall from earlier in this lesson that the direction of a vector is the counterclockwise angle of rotation that the vector makes with due East.)  

The Calculated Angle is Not Always the Direction

The measure of an angle as determined through use of SOH CAH TOA is not always the direction of the vector. The following vector addition diagram is an example of such a situation. Observe that the angle within the triangle is determined to be 26.6 degrees using SOH CAH TOA. This angle is the southward angle of rotation that the vector R makes with respect to West. Yet the direction of the vector as expressed with the CCW (counterclockwise from East) convention is 206.6 degrees.

tan(Theta) = (5/10) = 0.5

Theta = tan -1 (0.5)

Theta = 26.6 degrees

Direction of R = 90 deg + 26.6 deg

Direction of R = 116.6 deg

tan(Theta) = (40/30) = 1.333

Theta = tan -1 (1.333)

Theta = 53.1 degrees

Direction of R = 180 deg + 53.1 deg

Direction of R = 233.1 deg

In the above problems, the magnitude and direction of the sum of two vectors is determined using the Pythagorean theorem and trigonometric methods (SOH CAH TOA). The procedure is restricted to the addition of two vectors that make right angles to each other . When the two vectors that are to be added do not make right angles to one another, or when there are more than two vectors to add together, we will employ a method known as the head-to-tail vector addition method. This method is described below.

Use of Scaled Vector Diagrams to Determine a Resultant

physics vectors homework

The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position. Where the head of this first vector ends, the tail of the second vector begins (thus, head-to-tail method). The process is repeated for all vectors that are being added. Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to real units using the given scale. The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East.

A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below.

  • Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper.
  • Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m).
  • Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram.
  • Repeat steps 2 and 3 for all vectors that are to be added
  • Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as Resultant or simply R .
  • Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m).
  • Measure the direction of the resultant using the counterclockwise convention discussed earlier in this lesson .

An example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors:

The head-to-tail method is employed as described above and the resultant is determined (drawn in red). Its magnitude and direction is labeled on the diagram.

Interestingly enough, the order in which three vectors are added has no effect upon either the magnitude or the direction of the resultant. The resultant will still have the same magnitude and direction. For example, consider the addition of the same three vectors in a different order.

When added together in this different order, these same three vectors still produce a resultant with the same magnitude and direction as before (20. m, 312 degrees). The order in which vectors are added using the head-to-tail method is insignificant.

Additional examples of vector addition using the head-to-tail method are given on a separate web page .

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If you wanted to resolve the forces on an object, or find the resultant velocity, then you have to be able to add vectors together. Most of the time the two vectors you combine are at 90 degrees and you must be able to use trigonometry or scale drawing to find the result.

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physics vectors homework

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    Physics Problems and Solutions: Homework and Exam Physexams.com 1 Vectors 1.1 Unit Vectors 1. Find the unit vector in the direction w⃗= (5,2). Solution: A unit vector in physics is defined as a dimensionless vector whose magnitude is exactly 1. A unit vector that points in the direction of A⃗is determined by formula Aˆ = A⃗ |A⃗|

  2. 2: Vectors

    2.2: Scalars and Vectors (Part 1) Vectors are geometrically represented by arrows, with the end marked by an arrowhead. The length of the vector is its magnitude, which is a positive scalar. On a plane, the direction of a vector is given by the angle the vector makes with a reference direction, often an angle with the horizontal.

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    Includes 6 problems. Problem Set VP4 - Resolving Vectors into Components 2. Given the magnitude and direction of a vector, determine the magnitude and direction of its components. Includes 5 problems. Problem Set VP5 - Adding Two Perpendicular Vectors. Use Pythagorean theorem and trigonometric functions to determine the magnitude and direction ...

  4. 5.1 Vector Addition and Subtraction: Graphical Methods

    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.

  5. 3.2: Vectors

    A scalar is a physical quantity that can be represented by a single number. Unlike vectors, scalars do not have direction. Multiplying a vector by a scalar is the same as multiplying the vector's magnitude by the number represented by the scalar. Vectors are arrows consisting of a magnitude and a direction.

  6. PDF Chapter 3: Solutions of Homework Problems Vectors in Physics

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  11. Vector Addition

    Vector addition is one of the most common vector operations that a student of physics must master. When adding vectors, a head-to-tail method is employed. The head of the second vector is placed at the tail of the first vector and the head of the third vector is placed at the tail of the second vector; and so forth until all vectors have been added.

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