A football player runs the pattern given in the drawing by the three displacement vectors A, B, and C . The magnitudes of these vectors are A = 5 m, B = 16.0 m, and C = 16.0 m. Using the component method, find the magnitude and direction è of the resultant vector A,B,C. (Assume that up along the screen is the positive y-axis and that right is the positive x-axis.) The figure that is shown has vector A going up, B going from A's tip to the right (there's a 90 degree angle between A and B) then C going from B's tip to a 35 degree angle downward. A,B, and C are all connected be another vector.The theta that they want to find is at the tail of A.
I'd need to know the angular orientation of the A, B and C vectors to solve this. You have presumably been given that information. I suggest you learn the component method and apply it.
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To find the magnitude and direction of the resultant vector A,B,C, we can break down the vectors A, B, and C into their x and y components and then add them up.
First, let's calculate the x and y components of each vector:
Vector A:
- It goes up, which means it only has a y component.
- The magnitude of vector A is 5 m, so the y component Ay = 5 m.
Vector B:
- It goes from the tip of A to the right, forming a 90-degree angle with A.
- We can calculate the x and y components using trigonometry.
- The magnitude of vector B is 16.0 m, so we have:
- Bx = B * cos(90 degrees) = 0 (no x component)
- By = B * sin(90 degrees) = 16.0 m (y component)
Vector C:
- It goes from the tip of B to a 35-degree angle downward.
- Again, we can use trigonometry to find the x and y components.
- The magnitude of vector C is also 16.0 m, so we have:
- Cx = C * cos(35 degrees) (x component)
- Cy = -C * sin(35 degrees) (negative because it goes downward)
Now, let's add up the x and y components separately:
Rx = Ax + Bx + Cx
= 0 + 0 + (16.0 m * cos(35 degrees))
Ry = Ay + By + Cy
= 5 m + 16.0 m + (-16.0 m * sin(35 degrees))
Once you have the values for Rx and Ry, you can calculate the magnitude R of the resultant vector using the Pythagorean theorem:
R = sqrt(Rx^2 + Ry^2)
To find the direction è of the resultant vector, you can use trigonometry:
tan(è) = Ry / Rx
è = arctan(Ry / Rx)
Plug in the values for Rx and Ry, and calculate the magnitude R and direction è to find the final answer.