What is the resultant of this vector war?

Question

Fido pulls with a force of 525 Newtons at 160 degrees while Spot pulls with a force of 175 Newtons at 65 degrees and Rover pulls with a force of 825 Newtons at 315 degrees. What is the resultant force and degree?

Answer 1

The forces along the $x$- and $y$-direction are, respectively,

$$F_x=175\cos 65+525\cos 160 + 825\cos 315=163.98$$

$$F_y=175\sin 65+525\sin 160 + 825\sin 315=-245.20$$

The resulting force is

$$F= \sqrt{F_x^2+F_y^2} = 294.98 \>\text{N}$$

The resulting angle is in the 4th quadrant,

$$\theta = 360 +\arctan\frac {F_y}{F_x}=303.8 \>\text{degrees}$$