Conceptual Physics Reading and Study Workbook Chapter 5 Vector Addition and Resolution

Section Learning Objectives

By the stop of this section, yous volition be able to do the following:

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

Teacher Support

Instructor Support

The learning objectives in this section volition assist your students master the post-obit standards:

• (4) Science concepts. The student knows and applies the laws governing motion in two dimensions for a variety of situations. The student is expected to:
  • (E) develop and interpret gratuitous-trunk forcefulness diagrams.

Department Fundamental Terms

graphical method	head (of a vector)	head-to-tail method	resultant
resultant vector	tail	vector improver	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 directly-line move, the direction of a vector can be given simply by a plus or minus sign. Motion that is frontward, to the right, or upward is ordinarily considered to exist positive (+); and motion that is backward, to the left, or downward is usually considered to exist negative (−).

In 2 dimensions, a vector describes motility in two perpendicular directions, such every bit vertical and horizontal. For vertical and horizontal motion, each vector is made up of vertical and horizontal components. In a one-dimensional trouble, i of the components just has a value of naught. For two-dimensional vectors, we work with vectors by using a frame of reference such equally a coordinate arrangement. Simply every bit with 1-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.

Instructor Back up

Teacher Support

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

Figure v.2 shows a graphical representation of a vector; the total displacement for a person walking in a urban center. The person first walks nine blocks east and and then five blocks due north. Her total displacement does not friction match her path to her final destination. The deportation simply connects her starting bespeak with her ending point using a direct line, which is the shortest distance. Nosotros use the notation that a boldface symbol, such as D, stands for a vector. Its magnitude is represented past the symbol in italics, D, and its direction is given by an bending represented past the symbol $\theta .$ Annotation that her displacement would be the aforementioned if she had begun past first walking five blocks north and then walking nine blocks e.

Tips For Success

In this text, we stand for a vector with a boldface variable. For case, we represent a force with the vector F, which has both magnitude and management. The magnitude of the vector is represented past the variable in italics, F, and the direction of the variable is given by the bending $\theta .$

Figure v.2 A person walks nine blocks east and five blocks n. The deportation is 10.3 blocks at an angle ${\mathrm{29.i}}^{\circ }$ north of east.

The head-to-tail method is a graphical style 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. Allow the x-centrality represent the east-due west direction. Using a ruler and protractor, describe an arrow to correspond the start vector (nine blocks to the east), every bit shown in Figure v.iii(a).

   

   Figure 5.3 The diagram shows a vector with a magnitude of nine units and a management of 0°.

2. Let the y-axis represent the north-due south direction. Draw an arrow to represent the 2d vector (5 blocks to the north). Place the tail of the 2d vector at the head of the first vector, as shown in Figure 5.4(b).

   

   Figure v.4 A vertical vector is added.

3. If there are more than than ii vectors, go on to add the vectors head-to-tail as described in footstep 2. In this example, nosotros have only ii vectors, and so we have finished placing arrows tip to tail.
4. Describe an arrow from the tail of the first vector to the head of the last vector, as shown in Figure v.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 observe the magnitude of the resultant, measure its length with a ruler. When we bargain with vectors analytically in the side by side 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 bending it makes with the reference direction (in this case, the x-axis). When we deal with vectors analytically in the next department, the management will be calculated past using trigonometry to find the angle.

Teacher Support

Instructor Support

[AL] Ask two students to demonstrate pushing a table from 2 different directions. Ask students what they feel the management of resultant move will be. How would they represent this graphically? Think that a vector's magnitude is represented past the length of the arrow. Demonstrate the head-to-tail method of adding vectors, using the case given in the affiliate. Ask students to practice this method of add-on using a calibration and a protractor.

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

Sentinel Physics

Visualizing Vector Improver Examples

This video shows four graphical representations of vector add-on and matches them to the correct vector improver formula.

There are two vectors \text{a} and \text{b}. The head of vector \text{a} touches the tail of vector \text{b}. The addition of vectors \text{a} and \text{b} gives a resultant vector \text{c}. Can the addition of these two vectors can be represented by the post-obit two equations? \overrightarrow{\text{a}} + \overrightarrow{\text{b}} = \overrightarrow{\text{c}} ; \overrightarrow{\text{b}} + \overrightarrow{\text{a}} = \overrightarrow{\text{c}}

1. Yes, if we add the same two vectors in a dissimilar order it will however give the same resultant vector.

2. No, the resultant vector volition change if we add the same vectors in a different order.

Vector subtraction is done in the same manner every bit vector add-on 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 every bit the original vector, but points in the contrary direction (as shown in Figure 5.6). Subtracting the vector B from the vector A, which is written as A − B, is the aforementioned as A + (−B). Since information technology 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, v – two = 5 + (−2) = (−2) + 5.

Figure 5.vi 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 bending of ${\mathrm{thirty}}^{\circ }$ south of w is the same as the global angle ${210}^{\circ },$ which can also be expressed equally ${\mathrm{-150}}^{\circ }$ from the positive x-axis.

Using the Graphical Method of Vector Improver and Subtraction to Solve Physics Bug

At present that we have the skills to work with vectors in two dimensions, we tin can utilise vector improver to graphically determine the resultant vector, which represents the total force. Consider an example of force involving two ice skaters pushing a tertiary as seen in Figure v.seven.

Figure 5.7 Part (a) shows an overhead view of ii water ice skaters pushing on a third. Forces are vectors and add like vectors, then the total force on the tertiary skater is in the direction shown. In office (b), nosotros run into a free-trunk diagram representing the forces acting on the third skater.

In problems where variables such as forcefulness are already known, the forces can be represented past 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 North worth of force. In one case you have the initial vectors fatigued to calibration, yous tin and so utilize the head-to-tail method to describe the resultant vector. The length of the resultant can then be measured and converted dorsum to the original units using the scale you created.

Yous can tell by looking at the vectors in the free-trunk diagram in Effigy v.7 that the two skaters are pushing on the 3rd skater with equal-magnitude forces, since the length of their force vectors are the same. Note, however, that the forces are non equal considering they act in different directions. If, for example, each forcefulness had a magnitude of 400 N, then we would observe the magnitude of the full external force interim on the third skater past 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

$$\begin{array}{l}{a}^{\mathrm{two}}+b^2=c^2\hfill \\ c=\sqrt{a^2+b^2}.\hfill \end{array}$$

Applying this theorem to the triangle made by F ₁, F ₂, and F _tot in Figure 5.seven, we get

$$F_{\text{tot}}^{\mathrm{ii}}=\sqrt{F_1^2+F_1^2},$$

or

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

Note that, if the vectors were not at a right angle to each other $({90}^{\circ }$ to i some other), we would non be able to use the Pythagorean theorem to notice the magnitude of the resultant vector. Another scenario where adding 2-dimensional vectors is necessary is for velocity, where the direction may not be purely east-west or north-s, simply some combination of these 2 directions. In the next section, nosotros comprehend how to solve this type of problem analytically. For now let's consider the trouble graphically.

Worked Case

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

Utilize the graphical technique for adding vectors to find the full displacement of a person who walks the following 3 paths (displacements) on a flat field. First, she walks 25 one thousand in a direction ${49}^{\circ }$ north of east. Then, she walks 23 thou heading ${15}^{\circ }$ n of due 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 commencement A, the second B, and the third C. Brand the lengths proportional to the altitude of the given displacement and orient the arrows as specified relative to an east-west line. Use the head-to-tail method outlined in a higher place to make up one's mind the magnitude and direction of the resultant deportation, which we'll phone call R.

Give-and-take

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. Irresolute the order does not change the resultant. For instance, we could add the vectors every bit 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.

Teacher Support

Teacher Support

[BL] [OL] [AL] Ask iii students to enact the situation shown in Figure 5.viii. Retrieve how these forces can be represented in a free-body diagram. Giving values to these vectors, show how these can be added graphically.

Worked Example

Subtracting Vectors Graphically: A Woman Sailing a Boat

A adult female sailing a boat at night is following directions to a dock. The instructions read to first canvass 27.five m in a direction ${66.0}^{\circ }$ north of due east from her current location, and then travel 30.0 g in a management ${112}^{\circ }$ north of east (or ${22.0}^{\circ }$ westward of north). If the woman makes a mistake and travels in the reverse direction for the second leg of the trip, where volition she end upwards? The ii legs of the woman's trip are illustrated in Figure 5.13.

Figure 5.thirteen In the diagram, the first leg of the trip is represented past vector A and the second leg is represented past vector B.

Strategy

We can stand for the get-go leg of the trip with a vector A, and the second leg of the trip that she was supposed to have with a vector B. Since the woman mistakenly travels in the contrary direction for the second leg of the journey, the vector for second leg of the trip she really takes is −B. Therefore, she will stop up at a location A + (−B), or A − B. Note that −B has the same magnitude as B (30.0 m), just is in the contrary direction, ${68}^{\circ }({180}^{\circ }-{112}^{\circ })$ southward of due east, as illustrated in Figure 5.14.

Figure five.14 Vector –B represents traveling in the opposite direction of vector B.

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

Give-and-take

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

Worked Case

Adding Velocities: A Boat on a River

A gunkhole attempts to travel directly beyond a river at a speed of 3.8 m/s. The river current flows at a speed five _river of 6.1 k/s to the right. What is the total velocity and direction of the boat? Yous can represent each meter per 2d of velocity as one centimeter of vector length in your cartoon.

Strategy

We offset by choosing a coordinate system with its x-axis parallel to the velocity of the river. Because the boat is directed direct toward the other shore, its velocity is perpendicular to the velocity of the river. We describe the two vectors, v _boat and 5 _river, as shown in Figure 5.sixteen.

Using the caput-to-tail method, we draw the resulting total velocity vector from the tail of v _boat to the caput of v _river.

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

Give-and-take

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

Teacher Support

Teacher Back up

Teacher Sit-in

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

Do Bug

1 .

Vector \overrightarrow{\text{A}}, having magnitude ii.5\,\text{m}, pointing 37^\circ\! due south of eastward and vector \overrightarrow{\text{B}} having magnitude 3.5\,\text{m}, pointing 20^\circ\! northward of east are added. What is the magnitude of the resultant vector?

1. one.0\,\text{m}

2. 5.iii\,\text{m}

3. 5.9\,\text{m}

4. half dozen.0\,\text{m}

2 .

A person walks 32^\circ\! north of west for 94\,\text{1000} and 35^\circ\! east of s for 122\,\text{m}. What is the magnitude of his displacement?

1. 28\,\text{m}

2. 51\,\text{thousand}

3. 180\,\text{m}

4. 216\,\text{k}

Virtual Physics

Vector Addition

In this simulation, y'all will experiment with adding vectors graphically. Click and elevate the red vectors from the Take hold of One basket onto the graph in the centre of the screen. These red vectors tin can be rotated, stretched, or repositioned by clicking and dragging with your mouse. Check the Testify Sum box to display the resultant vector (in green), which is the sum of all of the ruby vectors placed on the graph. To remove a red vector, elevate it to the trash or click the Clear All button if you wish to start over. Notice that, if you click on whatever of the vectors, the $\left|R\right|$ is its magnitude, $\theta$ is its direction with respect to the positive 10-centrality, R₁₀ is its horizontal component, and R_y is its vertical component. You can check the resultant by lining up the vectors and so that the caput of the outset vector touches the tail of the second. Go along until all of the vectors are aligned together head-to-tail. You will see that the resultant magnitude and bending is the same equally the arrow drawn from the tail of the commencement vector to the head of the last vector. Rearrange the vectors in any club head-to-tail and compare. The resultant will ever be the aforementioned.

Grasp Cheque

True or Simulated—The more than long, red vectors y'all put on the graph, rotated in whatsoever direction, the greater the magnitude of the resultant green vector.

1. True
2. Faux

Check Your Understanding

3 .

While in that location is no single correct pick for the sign of axes, which of the following are conventionally considered positive?

1. backward and to the left

2. backward and to the right

3. forward and to the right

4. forward and to the left

4 .

Truthful or Simulated—A person walks two blocks east and 5 blocks northward. Some other person walks 5 blocks north and and so two blocks east. The displacement of the first person will be more than the deportation of the 2d person.

1. True
2. False

Instructor Support

Teacher Support

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

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Source: https://openstax.org/books/physics/pages/5-1-vector-addition-and-subtraction-graphical-methods