31 Introduction to Vectors and Vector Addition and Subtraction: Graphical Methods

OpenStax and Heath Hatch

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 2 shows such a graphical representation of a vector, using as an example the total displacement for the person walking in a city. We shall use the notation that a boldface symbol, such as , stands for a vector. Its magnitude is represented by the symbol in italics, and its direction by

VECTORS IN THIS TEXT

In this text, we will represent a vector with a vector symbol over the top variable. For example, we will represent the quantity force with the vector which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as and the direction of the variable will be given by an angle

The head-to-tail method is a graphical way to add vectors, described in Figure 4 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.

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

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.

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.

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.

Example 1: Adding Vectors Graphically Using the Head-to-Tail Method: A Women 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 25.0 m in a direction north of east. Then, she walks 23.0 m heading north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east.

Strategy

Represent each displacement vector graphically with an arrow, labeling the first the second and the third making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted

Solution

(1) Draw the three displacement vectors.

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

(3) Draw the resultant vector,

(4) Use a ruler to measure the magnitude of and a protractor to measure the direction of 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 the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector.

In this case, the total displacement is seen to have a magnitude of 50.0 m and to lie in a direction south of east. By using its magnitude and direction, this vector can be expressed as and 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 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 12 and we will still get the same solution.

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 or for example).

Vector Subtraction

Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract from written , we must first define what we mean by subtraction. The negative of a vector is defined to be that is, graphically the negative of any vector has the same magnitude but the opposite direction, as shown in Figure 13. In other words, has the same length as but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction.

The subtraction of vector from vector is then simply defined to be the addition of to Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results.

This is analogous to the subtraction of scalars (where, for example, ). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates.

Example 2: Subtracting Vectors Graphically: A Women 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 north of east from her current location, and then travel 30.0 m in a direction north of east (or 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? Compare this location with the location of the dock.

Strategy

We can represent the first leg of the trip with a vector and the second leg of the trip with a vector The dock is located at a location If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance (30.0 m) in the direction south of east. We represent this as as shown below. The vector has the same magnitude as but is in the opposite direction. Thus, she will end up at a location or

We will perform vector addition to compare the location of the dock, with the location at which the woman mistakenly arrives,

Solution

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

(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

In this case, and south of east.

(5) To determine the location of the dock, we repeat this method to add vectors and We obtain the resultant vector

In this case and north of east.

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 or 82.5 m, in a direction north of east. This is an example of multiplying a vector by a positive scalar. Notice that the magnitude changes, but the direction stays the same.

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 is multiplied by a scalar

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

In our case, and Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.

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.

Summary

• The graphical method of adding vectors and involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector is defined such that The magnitude and direction of are then determined with a ruler and protractor, respectively.
• The graphical method of subtracting vector from involves adding the opposite of vector which is defined as In this case, Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector
• Addition of vectors is commutative such that
• 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 is multiplied by a scalar quantity the magnitude of the product is given by If is positive, the direction of the product points in the same direction as if is negative, the direction of the product points in the opposite direction as

Problems & Exercises

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

1: Find the following for path A in Figure 21: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

2: Find the following for path B in Figure 21: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

3: Find the north and east components of the displacement for the hikers shown in Figure 19.

4: 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 and as in Figure 22, then this problem asks you to find their sum .)

5: Suppose you first walk 12.0 m in a direction west of north and then 20.0 m in a direction south of west. 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 and as in Figure 23, then this problem finds their sum .)

6: 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 which is 20.0 m in a direction exactly south of west, and then leg which is 12.0 m in a direction exactly west of north. (This problem shows that )

7: (a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction north of east (which is equivalent to subtracting from —that is, to finding ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction south of west and then 12.0 m in a direction east of south (which is equivalent to subtracting from —that is, to finding ). Show that this is the case.

8: Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors and all having different lengths and directions. Find the sum then find their sum when added in a different order and show the result is the same. (There are five other orders in which and can be added; choose only one.)

9: Show that the sum of the vectors discussed in Example 2 gives the result shown in Figure 17.

10: Find the magnitudes of velocities and in Figure 24

11: Find the components of along the x– and y-axes in Figure 24.

12: Find the components of along a set of perpendicular axes rotated counterclockwise relative to those in Figure 24.

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
the end point of a vector; the location of the tip of the vector’s arrowhead; also referred to as the “tip”
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

Solutions

Problems & Exercises

1:

(a)

(b) east of north

3:

north component 3.21 km, east component 3.83 km

5:

south of west

7:

(a) north of east

(b) south of west

9:

with respect to the x-axis.

11:

x-component 4.41 m/s

y-component 5.07 m/s

Homework Problems

12. Convert from a diagram to a vector-arithmetic expression.