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 = 13.0 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.)
To find the magnitude and direction of the resultant vector A + B + C, we will use the component method.
Step 1: Begin by drawing a coordinate system with the positive y-axis pointing up and the positive x-axis pointing right.
Step 2: Break down each vector (A, B, and C) into its x and y components using trigonometry.
For vector A:
- The magnitude of vector A is given as A = 5 m.
- The direction of vector A is not given in the question, so we assume it is an angle θ with respect to the positive x-axis.
- The x-component of vector A (Aₓ) is A * cos(θ).
- The y-component of vector A (Aᵧ) is A * sin(θ).
For vector B:
- The magnitude of vector B is given as B = 13.0 m.
- The direction of vector B is not given in the question, so we assume it is an angle φ with respect to the positive x-axis.
- The x-component of vector B (Bₓ) is B * cos(φ).
- The y-component of vector B (Bᵧ) is B * sin(φ).
For vector C:
- The magnitude of vector C is given as C = 16.0 m.
- The direction of vector C is not given in the question, so we assume it is an angle ψ with respect to the positive x-axis.
- The x-component of vector C (Cₓ) is C * cos(ψ).
- The y-component of vector C (Cᵧ) is C * sin(ψ).
Step 3: Add up the x-components and the y-components separately to find the resultant x-component (Rₓ) and resultant y-component (Rᵧ).
Rₓ = Aₓ + Bₓ + Cₓ
Rᵧ = Aᵧ + Bᵧ + Cᵧ
Step 4: Find the magnitude of the resultant vector (R) using the Pythagorean theorem:
R = √(Rₓ^2 + Rᵧ^2)
Step 5: Find the direction (θ) of the resultant vector:
θ = tan^(-1)(Rᵧ / Rₓ)
Step 6: Plug in the values and calculate the magnitude and direction of the resultant vector.