1.8

Addition of Vectors by Means of Components

The components of a vector provide the most convenient and accurate way of adding (or subtracting) any number of vectors. For example, suppose that vector is added to vector . The resultant vector is , where . Figure 1-21a illustrates this vector addition, along with the x and y vector components of and . In part b of the drawing, the vectors and have been removed, because we can use the vector components of these vectors in place of them. The vector component has been shifted downward and arranged tail to head with vector component . Similarly, the vector component has been shifted to the right and arranged tail to head with the vector component . The x components are colinear and add together to give the x component of the resultant vector . In like fashion, the y components are colinear and add together to give the y component of . In terms of scalar components, we can write

The vector components and of the resultant vector form the sides of the right triangle shown in Figure 1-21c. Thus, we can find the magnitude of by using the Pythagorean theorem:

The angle that makes with the x axis is given by . Example 9 illustrates how to add several vectors using the component method.

Figure 1-21 (a) The vectors and add together to give the resultant vector . The x and y components of and are also shown. (b) The drawing illustrates that and . (c) Vector and its components form a right triangle.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 9 | | | The Component Method of Vector Addition

A jogger runs 145 m in a direction 20.0° east of north (displacement vector ) and then 105 m in a direction 35.0° south of east (displacement vector ). Using components, determine the magnitude and direction of the resultant vector for these two displacements.
Reasoning Figure 1-22 shows the vectors and , assuming that the y axis corresponds to the direction due north. The vectors are arranged in a tail-to-head fashion, with the resultant vector drawn from the tail of to the head of . The components of the vectors are also shown in the figure. Since and its components form a right triangle (red in the drawing), we will use the Pythagorean theorem and trigonometry to express the magnitude and directional angle for in terms of its components. The components of will then be obtained from the components of and and the data given for these two vectors.

Figure 1-22 The vectors and add together to give the resultant vector . The vector components of and are also shown. The resultant vector can be obtained once its components have been found.

Knowns and Unknowns The data for this problem are listed in the table that follows:

Description | Symbol | Value | Comment
Magnitude of vector |  | 145 m |
Direction of vector |  | 20.0° east of north | see Figure 1-22.
Magnitude of vector |  | 105 m |
Direction of vector |  | 35.0° south of east | See Figure 1-22.
Unknown Variables |  |  |
Magnitude of resultant vector | C | ? |
Direction of resultant vector |  | ? |

Modeling the Problem

| Step 1 Magnitude and Direction of
| In Figure 1-22 the vector and its components and form a right triangle, as the red arrows show. Applying the Pythagorean theorem to this right triangle shows that the magnitude of is given by Equation 1a at the right. From the red triangle it also follows that the directional angle for the vector is given by Equation 1b at the right.

| (1a)

| (1b)

| Step 2 Components of
| Since vector is the resultant of vectors and , we have and can write the scalar components of as the sum of the scalar components of and :

These expressions can be substituted into Equations 1a and 1b for the magnitude and direction of , as shown at the right.

Solution Algebraically combining the results of each step, we find that

To use these results we need values for the individual components of and .
Referring to Figure 1-22, we find these values to be

Note that the component By is negative, because points downward, in the negative y direction in the drawing. Substituting these values into the results for C and gives

Related Homework: Problems 40, 41, 42, 43 and 50

Check Your Understanding 4

Two vectors, and , have vector components that are shown (to the same scale) in the drawing. Which vector (a, b, c, or d) is the vector sum of and ? (The answer is given at the end of the book.) Background: The concept of adding vectors by means of vector components is featured in this question. Note that the x components of and point in opposite directions, as do the y components.

For similar questions (including calculational counterparts), consult Self-Assessment Test 1.2. The test is described at the end of this section.
In later chapters we will often use the component method for vector addition. For future reference, the main features of the reasoning strategy used in this technique are summarized below.

| Reasoning Strategy

The Component Method of Vector Addition
1. | For each vector to be added, determine the x and y components relative to a conveniently chosen x, y coordinate system. Be sure to take into account the directions of the components by using plus and minus signs to denote whether the components point along the positive or negative axes.
2. | Find the algebraic sum of the x components, which is the x component of the resultant vector. Similarly, find the algebraic sum of the y components, which is the y component of the resultant vector.
3. | Use the x and y components of the resultant vector and the Pythagorean theorem to determine the magnitude of the resultant vector.
4. | Use either the inverse sine, inverse cosine, or inverse tangent function to find the angle that specifies the direction of the resultant vector.

Test your understanding of the concepts discussed in Sections 1.7 and 1.8:

· | The Vector and Scalar Components of a Vector
· | Adding Vectors by the Component Method

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