The three displacement vectors in the drawing have magnitudes of A = 5.48 m, B = 5.23 m, and C = 4.75 m. Find the resultant
((a) magnitude and (b) directional angle) of the three vectors by means of the component method. Express the directional angle as an angle above the positive or negative x axis.
angles are A=20.0 degrees and B=60.0 degrees
To find the resultant of the three vectors using the component method, we need to break down each vector into its x and y components.
Let's assume that vector A is represented as A_x in the x-direction and A_y in the y-direction. Similarly, vectors B and C can be represented as B_x, B_y, C_x, and C_y respectively.
Since the magnitudes of the vectors A, B, and C are given, we can calculate the components using the following formulas:
A_x = A * cos(θ)
A_y = A * sin(θ)
B_x = B * cos(θ)
B_y = B * sin(θ)
C_x = C * cos(θ)
C_y = C * sin(θ)
Next, we add up the x-components and the y-components separately:
R_x = A_x + B_x + C_x
R_y = A_y + B_y + C_y
To find the magnitude of the resultant vector, we use the Pythagorean theorem:
R = √(R_x² + R_y²)
To find the directional angle, we use the following formula:
θ = tan⁻¹(R_y / R_x)
Now, let's substitute the given values and calculate the resultant:
A = 5.48 m
B = 5.23 m
C = 4.75 m
First, calculate the components:
A_x = A * cos(θ) = 5.48 * cos(θ)
A_y = A * sin(θ) = 5.48 * sin(θ)
B_x = B * cos(θ) = 5.23 * cos(θ)
B_y = B * sin(θ) = 5.23 * sin(θ)
C_x = C * cos(θ) = 4.75 * cos(θ)
C_y = C * sin(θ) = 4.75 * sin(θ)
Next, calculate the resultant components:
R_x = A_x + B_x + C_x
R_y = A_y + B_y + C_y
Calculate the magnitude of the resultant vector:
R = √(R_x² + R_y²)
Finally, calculate the directional angle:
θ = tan⁻¹(R_y / R_x)
By substituting the values and calculating the above equations, you will find the magnitude and the directional angle of the resultant vector.