What is the magnitude of the third force?

Three forces with one unknown, find the magnitude and direction?

• Three cables pull on an object such that they create a resultant force having a magnitude of 1800N. If two of the forces pulling on the object are known determine the angle of the third cable so that the magnitude of force F in this cable is a minimum. All forces lie in the x-y plane. What is the magnitude of F? Force one is 1200N at 45 degrees above the x axis. Force two is 800N at 30 degrees below the x axis. Force three has an unknown force above the x axis at an unknown angle All x vectors are positive. Been stuck on this for a while now and I'm just not sure how to go about finding the unknown angle, if I knew how to get that I could solve this. I've broken the known forces into their constituent vectors. Force 1 = 848.5N for both x and y, both positive Force 2 = 692.8N to the right and 400N down The resulting magnitude of those is 1605N. That's as far as I've gotten.

• Answer:

  For the third force to be minimum, it should be in the same direction as that of the resultant of the two and its magnitude = 1800 - magnitude of the resultant of the two given forces. This means that we first find the resultant of the two forces and its direction. Sum of the horizontal and vertical components of the two are Rx = 1200cos45° + 800cos30° = 600√2 + 400√3 = 1541 N and Ry = 1200sin45° - 800sin30° = 600√2 + 400 = 1249 => Magnitude of the resultant = √[(1541)^2 + (1249)^2] = 1983.60 N Its direction makes an angle arctan(1249/1541) = 39° above x-axis => Third force should be of magnitude 183.60 N in the direction opposite to the resultant force which is 39° below negative x-axis.