A football player runs the pattern given in the drawing by the three displacement vectors A ,B ,C and . The magnitudes of these vectors are A = 4.00 m, B = 17.0 m, and C = 19.0 m. Using the component method, find the (a) magnitude and (b)direction of the resultant vector A + B + C . Take theta to be a positive angle.
I need to know the direction of ach vector.
To find the magnitude and direction of the resultant vector A + B + C using the component method, we need to break down each vector into its x and y components.
Let's assume that vector A is along the x-axis with a magnitude of 4.00 m. Therefore, its x-component (Ax) will be 4.00 m and its y-component (Ay) will be 0.
Next, let's assume that vector B makes an angle of 45 degrees with the positive x-axis and has a magnitude of 17.0 m. To find the x and y components of vector B, we can use the following equations:
Bx = B * cos(theta)
By = B * sin(theta)
Substituting the values, we get:
Bx = 17.0 m * cos(45 degrees) = 12.02 m
By = 17.0 m * sin(45 degrees) = 12.02 m
Now, let's assume that vector C makes an angle of 30 degrees with the positive x-axis and has a magnitude of 19.0 m. Using the same equations as above, we find:
Cx = C * cos(theta)
Cy = C * sin(theta)
Substituting the values, we get:
Cx = 19.0 m * cos(30 degrees) = 16.43 m
Cy = 19.0 m * sin(30 degrees) = 9.50 m
Now, add all the x-components and y-components separately to get the resultant vector:
Rx = Ax + Bx + Cx
Ry = Ay + By + Cy
Substituting the values, we get:
Rx = 4.00 m + 12.02 m + 16.43 m = 32.45 m
Ry = 0 + 12.02 m + 9.50 m = 21.52 m
To find the magnitude of the resultant vector, use the Pythagorean theorem:
R = sqrt(Rx^2 + Ry^2)
R = sqrt((32.45 m)^2 + (21.52 m)^2)
R ≈ 38.60 m (rounded to two decimal places)
To find the direction of the resultant vector, we can use the inverse tangent function:
theta = atan(Ry/Rx)
theta = atan(21.52 m / 32.45 m)
theta ≈ 33.15 degrees (rounded to two decimal places)
Since theta was defined as a positive angle, the direction of the resultant vector A + B + C is 33.15 degrees.