1.9

Concepts & Calculations

This chapter has presented an introduction to the mathematics of trigonometry and vectors, which will be used throughout this text. Therefore, in this last section we consider several examples in order to review some of the important features of this mathematics. The three-part format of these examples stresses the role of conceptual understanding in problem solving. First, the problem statement is given. Then, there is a concept question-and-answer section, which is followed by the solution section. The purpose of the concept question-and-answer section is to provide help in understanding the solution and to illustrate how a review of the concepts can help in anticipating some of the characteristics of the numerical answers.

| Concepts & Calculations Example 10 | | | Equal Vectors

Figure 1-23 shows two displacement vectors and . Vector points at an angle of 22.0° above the x axis and has an unknown magnitude. Vector has an x component and has an unknown y component By. These two vectors are equal. Find the magnitude of and the value of By.

Figure 1-23 The two displacement vectors and are equal. Example 10 discusses what this equality means.

Concept Questions and Answers What does the fact that vector equals vector imply about the magnitudes and directions of the vectors?

Answer When two vectors are equal, each has the same magnitude and each has the same direction.

What does the fact that vector equals vector imply about the x and y components of the vectors?

Answer When two vectors are equal, the x component of vector equals the x component of vector and the y component of vector equals the y component of vector .

Solution We focus on the fact that the x components of the vectors are the same and the y components of the vectors are the same. This allows us to write that

| (1.8)

| (1.9)

Dividing Equation 1.9 by Equation 1.8 shows that

Solving Equation 1.8 directly for A gives

| Concepts & Calculations Example 11 | | | Using Components to Add Vectors

Figure 1-24 shows three displacement vectors , , and . These vectors are arranged in tail-to-head fashion, because they add together to give a resultant displacement , which lies along the x axis. Note that the vector is parallel to the x axis. What is the magnitude of the vector ?

Figure 1-24 The displacement vectors , , and add together to give the resultant displacement , which lies along the x axis. Note that the vector is parallel to the x axis.

Concept Questions and Answers How is the magnitude of related to its scalar components Cx and Cy?

Answer The magnitude of is given by the Pythagorean theorem in the form , since a vector and its components form a right triangle (see Figure 1-16).

Do any of the vectors in Figure 1-24 have a zero value for either their x or y components?

Answer Yes. The vectors and each have a zero value for their y component. This is because these vectors are parallel to the x axis.

What does the fact that , , and add together to give the resultant tell you about the components of these vectors?

Answer The fact that means that the sum of the x components of , , and equals the x component of and the sum of the y components of , , and equals the y component of .

Solution To begin with, we apply the Pythagorean theorem to relate C, the magnitude of the vector , to its scalar components Cx and Cy;

To obtain a value for Cx we use the fact that the sum of the x components of , , and equals the x component of :

To obtain a value for Cy we note that the sum of the y components of , , and equals the y component of (and remember that and have no y components):

With these values for Cx and Cy we find that the magnitude of is

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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