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 A is added to vector B. The resultant vector is C, where C=A+B. Figure 1.21a illustrates this vector addition, along with the x and y vector components of A and B. In part b of the drawing, the vectors A and B have been removed, because we can use the vector components of these vectors in place of them. The vector component B_x has been shifted downward and arranged tail to head with the vector component A_x. Similarly, the vector component A_y has been shifted to the right and arranged tail to head with the vector component B_y. The x components are colinear and add together to give the x component of the resultant vector C. In like fashion, the y components are colinear and add together to give the y component of C. In terms of scalar components, we can write

The vector components C_x and C_y of the resultant vector form the sides of the right triangle shown in Figure 1.21c. Thus, we can find the magnitude of C by using the Pythagorean theorem:

The angle q  that C makes with the x axis is given by q = tan^–1 (C_y/C_x). Example 8 illustrates how to add several vectors using the component method.

Figure 1.21  (a) The vectors A and B add together to give the resultant vector C. The x and y vector components of A and B are also shown. (b) The drawing illustrates that C x = A x + B x and C y = A y + B y . (c) Vector C and its components form a right triangle.

A jogger runs 145 m in a direction 20.0° east of north (displacement vector A) and then 105 m in a direction 35.0° south of east (displacement vector B). Determine the magnitude and direction of the resultant vector C for these two displacements. Reasoning  Figure 1.22a shows the vectors A and B, assuming that the y axis corresponds to the direction due north. Since the vectors are not given in component form, we will begin by using the given magnitudes and directions to find the components. Then, the components of A and B can be used to find the components of the resultant C. Finally, with the aid of the Pythagorean theorem and trigonometry, the components of C can be used to find its magnitude and direction. Figure 1.22  (a) The vectors A and B add together to give the resultant vector C. The vector components of A and B are also shown. (b) The resultant vector C can be obtained once its components have been found. Solution The first two rows of the following table give the x and y components of the vectors A and B. Note that the component By is negative, because By points downward, in the negative y direction in the drawing.  Vector   x Component   y Component   A         B         C        The third row in the table gives the x and y components of the resultant vector C: Cx=Ax+Bx and Cy=Ay+By. Part b of the drawing shows C and its vector components. The magnitude of C is given by the Pythagorean theorem as The angle q  that C makes with the x axis is

Two vectors, A and B, have vector components that are shown (to the same scale) in the first row of drawings. Which vector R in the second row of drawings is the vector sum of A and B? The concept of adding vectors by means of vector components is featured in this question. Note that the x components of A and B 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.

Concept Simulation 1.1

This simulation illustrates how two vectors can be added by the tail-to-head method and by the component method. The user can change the magnitude and direction of each vector, and the simulation shows how their components, as well as those of the resultant vector, change. Related Homework: Problems 40, 45, 55

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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