# 3.2: Vector Addition and Subtraction: Graphical Methods

## Vector Addition and Subtraction: Graphical Methods

Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai’i to Moloka’i has a number of legs, or journey segments. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. (credit: US Geological Survey)

### Vectors in Two Dimensions

A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, 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. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

Figure shows such a *graphical representation of a vector*, using as an example the total displacement for the person walking in a city considered in Kinematics in Two Dimensions: An Introduction. We shall use the notation that a boldface symbol, such as

### VECTORS IN THIS TEXT

In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector

*A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle **29.1º** north of east. *

*To describe the resultant vector for the person walking in a city considered in Figure graphically, draw an arrow to represent the total displacement vector **D**. Using a protractor, draw a line at an angle **θ** relative to the east-west axis. The length **D** of the arrow is proportional to the vector’s magnitude and is measured along the line with a ruler. In this example, the magnitude **D** of the vector is 10.3 units, and the direction **θ** is **29.1º** north of east.*

### Vector Addition: Head-to-Tail Method

The head-to-tail method is a graphical way to add vectors, described in Figure below and in the steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.

*Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in Figure. (a) Draw a vector representing the displacement to the east. (b) Draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first, east-pointing vector. (c) Draw a line from the tail of the east-pointing vector to the head of the north-pointing vector to form the sum or resultant vector **D**. The length of the arrow **D** is proportional to the vector’s magnitude and is measured to be 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) **θ** is measured with a protractor to be **29.1º**. *

*Step 1.* *Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor*.

</figure>

*Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail of the second vector at the head of the first vector.*

</figure>

*Step 3.* *If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail*.

*Step 4.* *Draw an arrow from the tail of the first vector to the head of the last vector*. This is the resultant, or the sum, of the other vectors.

</figure>

*Step 5.* To get the magnitude of the resultant, *measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.)*

*Step 6. *To get the direction of the resultant, *measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.)*

The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors.

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

Strategy

Represent each displacement vector graphically with an arrow, labeling the first

Solution

(1) Draw the three displacement vectors.

<figure id="import-auto-id1165296232338" style="width: 810px;"></figure>

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

<figure id="import-auto-id1165298788198" style="width: 810px;"></figure>

(3) Draw the resultant vector,

</figure>

(4) Use a ruler to measure the magnitude of

</figure>

In this case, the total displacement

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 and we will still get the same solution.

<figure id="import-auto-id1165298931858" style="width: 810px;"></figure>

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

# Vector Subtraction

Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract *negative* of a vector *the negative of any vector has the same magnitude but the opposite direction*, as shown in Figure. In other words,

<figcaption>

The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So

The *subtraction* of vector

This is analogous to the subtraction of scalars (where, for example,

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 *opposite* direction for the second leg of the trip, where will she end up? Compare this location with the location of the dock.

</figure>

Strategy

We can represent the first leg of the trip with a vector *opposite* direction for the second leg of the journey, she will travel a distance

</figure>

We will perform vector addition to compare the location of the dock,

Solution

(1) To determine the location at which the woman arrives by accident, draw vectors

(2) Place the vectors head to tail.

(3) Draw the resultant vector

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

</figure>

In this case,

(5) To determine the location of the dock, we repeat this method to add vectors

</figure>

In this case

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.

# Multiplication of Vectors and Scalars

If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk

If the scalar is negative, then multiplying a vector by it changes the vector’s magnitude and gives the new vector the *opposite*direction. For example, if you multiply by –2, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector

- the magnitude of the vector becomes the absolute value of
c A , - if
c is positive, the direction of the vector does not change, - if
c is negative, the direction is reversed.

In our case,

# Resolving a Vector into Components

In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular components of a single vector, for example the *x*-* and* *y*-components, or the north-south and east-west components.

For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction *finding the components (or parts)* of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in Projectile Motion, and much more when we cover forces in Dynamics: Newton’s Laws of Motion. Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in Vector Addition and Subtraction: Analytical Methods are ideal for finding vector components.

Learn about position, velocity, and acceleration in the "Arena of Pain". Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can.

<figure class="ui-has-child-figcaption" id="eip-id1434453" style="width: 660px;"> <figcaption># Summary

- The graphical method of adding vectors
A andB involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vectorR is defined such thatA+B=R . The magnitude and direction ofR are then determined with a ruler and protractor, respectively. - The graphical method of subtracting vector
B fromA involves adding the opposite of vectorB , which is defined as−B . In this case,A–B=A+(–B)=R . Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vectorR . - Addition of vectors is commutative such that
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.
- If a vector
A is multiplied by a scalar quantityc , the magnitude of the product is given bycA . Ifc is positive, the direction of the product points in the same direction asA ; ifc is negative, the direction of the product points in the opposite direction asA .

# Conceptual Questions

Which of the following is a vector: a person’s height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this book, the Earth’s population, the acceleration of gravity?

Give a specific example of a vector, stating its magnitude, units, and direction.

What do vectors and scalars have in common? How do they differ?

Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?

<figure id="import-auto-id1165298840401" style="width: 782px;"></figure>

If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in Figure. What other information would he need to get to Sacramento?

<figure id="import-auto-id1165296384452" style="width: 782px;"></figure>

Suppose you take two steps

Explain why it is not possible to add a scalar to a vector.

If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

# Problems & Exercises

Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.

Find the following for path A in Figure: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

<figure class="ui-has-child-figcaption" id="import-auto-id1165298872310" style="width: 782px;"><figcaption>

The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

(a)

(b)

Find the following for path B in Figure: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

Find the north and east components of the displacement for the hikers shown in Figure.

north component 3.21 km, east component 3.83 km

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements

<figcaption>

The two displacements

Suppose you first walk 12.0 m in a direction

</figure>

Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg

(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction

(a)

(b)

Show that the *order* of addition of three vectors does not affect their sum. Show this property by choosing any three vectors

*x*-axis.

Find the magnitudes of velocities

<figcaption>

The two velocities

Find the components of *x*- and *y*-axes in Figure.

*x*-component 4.41 m/s

*y*-component 5.07 m/s

Find the components of

## Glossary

- component (of a 2-d vector)
- a piece of a vector that points in either the vertical or the horizontal direction; every 2-d vector can be expressed as a sum of two vertical and horizontal vector components

- commutative
- refers to the interchangeability of order in a function; vector addition is commutative because the order in which vectors are added together does not affect the final sum

- direction (of a vector)
- the orientation of a vector in space

- head (of a vector)
- the end point of a vector; the location of the tip of the vector’s arrowhead; 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

- magnitude (of a vector)
- the length or size of a vector; magnitude is a scalar quantity

- resultant
- the sum of two or more vectors

- resultant vector
- the vector sum of two or more vectors

- scalar
- a quantity with magnitude but no direction

- tail
- the start point of a vector; opposite to the head or tip of the arrow

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