Vector C has a magnitude 25.8 m and is in the direction of the negative y axis. Vectors A and B are at angles á = 41.9° and â = 26.7° up from the x axis respectively. If the vector sum A B C = 0, what are the magnitudes of A and B?
A = a [cos(41.9°), sin(41.9°)]
B = b [cos(26.7°), sin(26.7°)]
C = 25.8 [0, -1]
A+B+C = [0,0]
0 = a cos(41.9°) + b cos(26.7°)
0 = a sin(41.9°) + c sin(26.7°) - 1
Solve for a and b
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To find the magnitudes of vectors A and B, we can start by analyzing the given information. We know that the vector sum of A, B, and C is zero, which means that the combined effect of these vectors cancels out.
1. Before proceeding, let's define the coordinate axes for clarity:
- The x-axis is the horizontal axis
- The y-axis is the vertical axis
2. Vector C has a magnitude of 25.8 m and is in the direction of the negative y-axis. This means vector C points downward and has a negative y-component.
3. Vector A is at an angle of á = 41.9° up from the x-axis. To find the components of vector A, we can use trigonometry. Split the vector A into its x and y components as follows:
- Ax = A * cos(á)
- Ay = A * sin(á)
Here, A refers to the magnitude of vector A.
4. Vector B is at an angle of â = 26.7° up from the x-axis. Similar to vector A, find the x and y components of vector B using trigonometry:
- Bx = B * cos(â)
- By = B * sin(â)
Here, B refers to the magnitude of vector B.
5. Since the vector sum A + B + C = 0, we can combine the x and y components of the vectors (Ax + Bx = 0 and Ay + By + Cy = 0). Since the vectors cancel each other out, the sum of their components must equal zero.
6. Let's write the equations for the vector sum in terms of components:
- Ax + Bx = 0 (Equation 1)
- Ay + By + Cy = 0 (Equation 2)
7. From Equation 1, we have Ax = -Bx. By substituting the equations for Ax and Bx from steps 3 and 4, we get:
- A * cos(á) = -B * cos(â)
8. From Equation 2, we have Ay + By + Cy = 0. Substituting the equations for Ay, By, and Cy from earlier, we get:
- A * sin(á) + B * sin(â) + (-25.8) = 0
9. We have two equations (one from step 7 and one from step 8) with two unknowns (A and B). By solving these equations simultaneously, we can find the magnitudes of vectors A and B.
10. Substitute the angles (á = 41.9° and â = 26.7°) and the known magnitude of vector C (25.8 m) into the equations from steps 7 and 8.
11. Solve the resulting system of equations to find the values of A and B that satisfy these equations.