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 = 5.0 m, B= 15 m, and C = 18 m. Using the component method, find the magnitude and direction è of the resultant vector
You have made this problem impossible by the most fundamental error.
You gave us scalars, not vectors!
Vectors have both magnitude (scalar length) and DIRECTION !!!
Without direction, the three "things" are simply not vectors and they certainly do not have components.
To find the magnitude and direction of the resultant vector, we can use the component method. The component method involves breaking down the vectors into their horizontal and vertical components.
Step 1: Draw a coordinate system with x and y axes, and label them accordingly.
Step 2: Calculate the horizontal (x-direction) and vertical (y-direction) components of each vector.
For vector A:
- The angle of vector A with the x-axis is not given, so let's assume it is 0 degrees (along the positive x-axis).
- The horizontal component of vector A (Ax) is A * cos(0) = 5 * cos(0) = 5.
- The vertical component of vector A (Ay) is A * sin(0) = 5 * sin(0) = 0.
For vector B:
- The angle of vector B with the x-axis is given in the drawing as 60 degrees.
- The horizontal component of vector B (Bx) is B * cos(60) = 15 * cos(60) = 7.5.
- The vertical component of vector B (By) is B * sin(60) = 15 * sin(60) = 12.99 (approximately).
For vector C:
- The angle of vector C with the x-axis is not given, so let's assume it is 0 degrees (along the positive x-axis).
- The horizontal component of vector C (Cx) is C * cos(0) = 18 * cos(0) = 18.
- The vertical component of vector C (Cy) is C * sin(0) = 18 * sin(0) = 0.
Step 3: Add up the horizontal components and the vertical components separately.
Horizontal component (Rx) = Ax + Bx + Cx = 5 + 7.5 + 18 = 30.5.
Vertical component (Ry) = Ay + By + Cy = 0 + 12.99 + 0 = 12.99 (approximately).
Step 4: Use the Pythagorean theorem to find the magnitude of the resultant vector.
Magnitude (R) = sqrt(Rx^2 + Ry^2) = sqrt(30.5^2 + 12.99^2) = sqrt(930.25 + 168.44) = sqrt(1098.69) = 33.14 (approximately).
Step 5: Calculate the direction (θ) of the resultant vector.
θ = atan(Ry / Rx) = atan(12.99 / 30.5) = atan(0.43) = 23.25 degrees (approximately).
Therefore, the magnitude of the resultant vector is 33.14 m, and the direction is 23.25 degrees.