If the magnitude of the resultant force is to be 450 N , directed along the positive y axis, determine the magnitude of force F.

Determine the force F direction θ.

well, the projection of u onto v is u•v/|v|

To determine the magnitude of force F, we can use the concept of vector addition.

Here's how you can solve it step by step:

Step 1: Start by labeling the given information. In the image, the angles of the forces are provided as θ = 30° and φ = 45°. The magnitude of the resultant force is given as 450 N, directed along the positive y-axis.

Step 2: Break down the forces into their x and y components. In this case, force F has components Fx and Fy, while force P has components Px and Py.

Step 3: Use trigonometry to find the x and y components of each force. For example, for force F, we can use the given angle θ to find Fx = F * cos(θ) and Fy = F * sin(θ). Similarly, for force P, we can use the given angle φ to find Px = P * cos(φ) and Py = P * sin(φ).

Step 4: Add up the x and y components of both forces separately. This will give us the x and y components of the resultant force. Let's call them Rx and Ry.

Rx = Fx + Px
Ry = Fy + Py

Step 5: Use the Pythagorean theorem to find the magnitude of the resultant force:

Magnitude of the resultant force (R) = sqrt(Rx^2 + Ry^2)

Step 6: Set the magnitude of the resultant force equal to 450 N (as given in the question), and solve the equation for the magnitude of force F.

450 = sqrt(Rx^2 + Ry^2)

Step 7: Plug in the known values for Rx and Ry, and solve for F.

450 = sqrt((Fx + Px)^2 + (Fy + Py)^2)

By substituting the given angle values for θ and φ, you can now calculate the magnitude of force F using algebraic manipulation and arithmetic calculations.