y x z Forces => Vectors => 2D and 3D Force System = 2 or more forces acting on a body (particle or mass) Resultant =  forces in the system R = F = F1 + F2 + F3 where R = Rxî + Ryĵ + Rzk Let’s look at three random forces and find the resultant analytically and graphically. x z y

F1 = 2N 30 1st quadrant F2 = 4N 150 2nd quadrant F3 = 6N -45 = 6N 315 4th quadrant F1 = 2N 30 = 2 cos 30 î + 2 sin 30 ĵ = 1.732 î + 1ĵ F2 = 4N 150 = 4 cos 150 î + 4 sin 150 ĵ = (-)3.464 î + 2 ĵ F3 = 6N 315 = 6 cos 315 î + 6 sin 315 ĵ = 4.243 î + (-)4.243 ĵ

Add like terms of these forces to get the resultant, R = F = F1 + F2 + F3 F1 = 1.732 î + 1 ĵ + F2 = (-)3.464 î + 2 ĵ + F3 = 4.243 î + (-)4.243 ĵ R = 2.511 î + (-)1.243 ĵ R = Rx î + Ry ĵ Convert resultant to polar coordinates. magnitude =  (2.511)² + (-1.243)² = 2.802  2.8 N orientation = tan-1 (rise / run) = tan-1 (-1.243/2.511) = -26.3  R = 2.8 N -26.3  2.8 N 334