A force of 50N acts in an easterly direction on an object at the same time as a force of 80N pushes on it in the direction of N 45 degrees E. Determine the magnitude and direction of resultant force

Write each force in component form:

v ₁ : 50 N due east   →   (50 N) i

v ₂ : 80 N at N 45° E   →   (80 N) (cos(45°) i + sin(45°) j ) ≈ (56.5 N) (i + j )

The resultant force is the sum of these two vectors:

r = v ₁ + v ₂ ≈ (106.5 N) i + (56.5 N) j

Its magnitude is

|| r || = √[(106.5 N)² + (56.5 N)²] ≈ 121 N

and has direction θ such that

tan(θ) = (56.5 N) / (106.5 N)   →   θ ≈ 28.0°

i.e. a direction of about E 28.0° N. (Just to clear up any confusion, I mean 28.0° north of east, or 28.0° relative to the positive x-axis.)