Before starting this problem, review Conceptual Example 7. The force vector A has a magnitude of 70.0 newtons (N) and points due east. The force vector B has a magnitude of 111 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)

Question

Before starting this problem, review Conceptual Example 7. The force vector A has a magnitude of 70.0 newtons (N) and points due east. The force vector B has a magnitude of 111 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)

Answer 1

To solve this problem, we can break down the given forces into their x-components and y-components. Then we can perform vector addition and subtraction to find the resulting vector.

Let's start by finding the components for each force:

For force A:
- Magnitude: 70.0 N (given)
- Direction: Due east

Since the force is due east, it means that its x-component will be positive (in the positive x-direction) and the y-component will be zero.

Ax = 70.0 N (x-component of A)
Ay = 0 N (y-component of A)

For force B:
- Magnitude: 111 N (given)
- Direction: 71.4 ° north of east

To find the components of force B, we need to resolve it into its x- and y-components.

Since B is 71.4 ° north of east, we can calculate the components using trigonometry. The angle is measured from the positive x-axis counterclockwise.

Bx = 111 N * cos(71.4 °)
By = 111 N * sin(71.4 °)

Now that we have the components, we can proceed to find the resulting vectors:

(a) To find the magnitude of A - B:
The magnitude of the resulting vector A - B can be found using the Pythagorean theorem:

|A - B| = sqrt((Ax - Bx)^2 + (Ay - By)^2)

(b) To find the direction of A - B:
The direction of the resulting vector can be calculated using the inverse tangent function (arctan) applied to the ratio of the y-component to the x-component:

θ = arctan((Ay - By) / (Ax - Bx))

(c) To find the magnitude of B - A:
The magnitude of the resulting vector B - A can be found in the same way as above:

|B - A| = sqrt((Bx - Ax)^2 + (By - Ay)^2)

(d) To find the direction of B - A:
The direction of the resulting vector can be calculated using the inverse tangent function (arctan) applied to the ratio of the y-component to the x-component:

θ = arctan((By - Ay) / (Bx - Ax)).

Now you have all the steps needed to solve the problem. Substitute the given values into the equations, perform the calculations, and you will get the answers to each part.