Resolve each force into its x and y components.

To find the resultant force, we need to resolve each force into its x and y components, sum these components, and then calculate the magnitude and direction of the resultant vector. Step 1: Resolve each force into its x and y components. Force 1: F_1 = 6\,N (acting vertically upwards, along the positive y-axis) F_1x = 6 (90^) = 0\,N F_1y = 6 (90^) = 6\,N Force 2: F_2 = 10\,N (acting at 30^ above the positive x-axis) F_2x = 10 (30^) = 10 (sqrt(3)2) = 5sqrt(3)\,N F_2y = 10 (30^) = 10 ((1)/(2)) = 5\,N Force 3: F_3 = 3\,N (acting horizontally, along the positive x-axis) F_3x = 3 (0^) = 3\,N F_3y = 3 (0^) = 0\,N Force 4: F_4 = 3\,N (acting at 60^ below the negative x-axis) This force is in the third quadrant. Both x and y components will be negative. F_4x = -3 (60^) = -3 ((1)/(2)) = -1.5\,N F_4y = -3 (60^) = -3 (sqrt(3)2) = -1.5sqrt(3)\,N Step 2: Sum the x-components to find the total x-component (R_x). R_x = F_1x + F_2x + F_3x + F_4x R_x = 0 + 5sqrt(3) + 3 - 1.5 R_x = 5sqrt(3) + 1.5\,N Numerically: R_x ≈ 5(1.732) + 1.5 = 8.66 + 1.5 = 10.16\,N Step 3: Sum the y-components to find the total y-component (R_y). R_y = F_1y + F_2y + F_3y + F_4y R_y = 6 + 5 + 0 - 1.5sqrt(3) R_y = 11 - 1.5sqrt(3)\,N Numerically: R_y ≈ 11 - 1.5(1.732) = 11 - 2.598 = 8.402\,N **Step 4: Calculate the magnitude of the resultant