Before starting this problem, review Conceptual Example 7. The force vector A has a magnitude of 97.0 newtons (N) and points due east. The force vector B has a magnitude of 107 N and points 71.4 ° north of east. Find the (a) magnitude and (b) direction of A - B. (Give the direction as a positive angle with respect to due east). Find the (c) magnitude and (d) direction of B - A. (Give the direction as a positive angle with respect to due west).
F = 97N - 107N[71.4]
Fx = 97 - 107*Cos71.4 = 62.87 N.
Fy = 0 - 107*sin71.4 = -101.4 N.
a. F^2 = Fx^2 + Fy^2=62.87^2 + (-101.4^2) = 119.3 N.
F = N.
b. Tan A = Fy/Fx = -101.4/62.87 = -1.61285
A = -58.2o S of E = 301.8o CCW
c. 107N[71.4o] - 97
To find the magnitude and direction of A - B and B - A, we can use vector addition and subtract the two vectors in the given order.
To find the magnitude of A - B:
1. Draw the vector A with a magnitude of 97.0 N pointing due east.
2. Draw the vector B with a magnitude of 107 N pointing 71.4° north of east.
3. To subtract B from A, we can think of it as adding the negative of vector B to vector A.
4. Find the negative of vector B by changing its direction by 180°, resulting in a vector with a magnitude of 107 N pointing 71.4° south of east.
5. Now, we can add vector A and the negative of vector B by placing the tail of the negative B at the head of vector A.
6. Draw the resultant vector, which is the vector A - B, from the tail of vector A to the head of the negative B.
7. Measure the magnitude of the resultant vector, A - B, using a ruler or measuring tool.
To find the direction of A - B:
8. Measure the angle between the resultant vector, A - B, and the positive x-axis (east) using a protractor or angle measuring tool.
9. Note the direction as a positive angle with respect to due east.
To find the magnitude of B - A:
10. Similarly, draw the vector B with a magnitude of 107 N pointing 71.4° north of east.
11. Draw the vector A with a magnitude of 97.0 N pointing due east.
12. To subtract A from B, we can think of it as adding the negative of vector A to vector B.
13. Find the negative of vector A by changing its direction by 180°, resulting in a vector with a magnitude of 97.0 N pointing due west.
14. Now, we can add vector B and the negative of vector A by placing the tail of the negative A at the head of vector B.
15. Draw the resultant vector, which is the vector B - A, from the tail of vector B to the head of the negative A.
16. Measure the magnitude of the resultant vector, B - A, using a ruler or measuring tool.
To find the direction of B - A:
17. Measure the angle between the resultant vector, B - A, and the positive x-axis (west) using a protractor or angle measuring tool.
18. Note the direction as a positive angle with respect to due west.
Once you have performed these steps to obtain the magnitudes and directions, you can report the results as (a) magnitude and (b) direction of A - B, and (c) magnitude and (d) direction of B - A.