Three forces with magnitudes 75 pound, 100 lb, and 125 pound act on an object at angles of 30 degrees, -45 degrees, and 120 degrees respectively. Find the direction and magnitude of the resultant vector.

To find the magnitude and direction of the resultant vector, we need to use vector addition.

Step 1: Resolve each force into its horizontal (x) and vertical (y) components.
To resolve a force into its components, we use the following equations:
Fx = F * cos(θ)
Fy = F * sin(θ)

For the first force with magnitude 75 lb and an angle of 30 degrees:
F1x = 75 lb * cos(30) = 64.95 lb
F1y = 75 lb * sin(30) = 37.5 lb

For the second force with magnitude 100 lb and an angle of -45 degrees:
F2x = 100 lb * cos(-45) = 70.71 lb
F2y = 100 lb * sin(-45) = -70.71 lb

For the third force with magnitude 125 lb and an angle of 120 degrees:
F3x = 125 lb * cos(120) = -62.5 lb
F3y = 125 lb * sin(120) = 107.78 lb

Step 2: Calculate the sum of the x-components and y-components of the forces.
To find the resultant vector, we sum the x-components and the y-components separately.

F_res_x = F1x + F2x + F3x = 64.95 lb + 70.71 lb - 62.5 lb = 73.16 lb
F_res_y = F1y + F2y + F3y = 37.5 lb - 70.71 lb + 107.78 lb = 74.57 lb

Step 3: Calculate the magnitude of the resultant vector.
To find the magnitude of the resultant vector, we use the Pythagorean theorem:
|F_res| = √(F_res_x^2 + F_res_y^2)

|F_res| = √(73.16 lb^2 + 74.57 lb^2) = √(5348.99 lb^2) = 73.1 lb (rounded to one decimal place)

Step 4: Calculate the direction of the resultant vector.
To find the direction, we use the inverse tangent (arctan) function:
θ_res = arctan(F_res_y / F_res_x)

θ_res = arctan(74.57 lb / 73.16 lb) ≈ 45.3 degrees (rounded to one decimal place)

Therefore, the magnitude of the resultant vector is 73.1 lb, and its direction is approximately 45.3 degrees.