Two tugboats push a barge, one with 2.7 x 10^5 N along the x direction and one with 3.6 x 10^5 N along the y direction. What is the magnitude of the total applied force?

1. The two applied forces are perpendicular to each other. 2. The magnitude of the resultant of perpendicular vectors comes from the Pythagorean theorem. 3. Square each force: (2.7 x 10^5 N)^2 = 7.29 x 10^10 and (3.6 x 10^5 N)^2 = 1.296 x 10^11. 4. Add them: 2.025 x 10^11 N^2. 5. Take the square root: 4.5 x 10^5 N. 6. Simply adding the two magnitudes gives 6.3 x 10^5 N, which would only be right if they were parallel. _Source: OpenStax College Physics (CC BY 4.0), Ch 4 "Dynamics: Force and Newton's Laws of Motion", section 4.7 Further Applications of Newton's Laws of Motion_