Find resultant magnitude and direction if a force of 14N at 66 degrees above due east and a force of 27N at 19 degrees below due west act on an object.

Answer 1

To add vectors that are in different directions, you can break each vector down into its components (in the x and y direction or in this case, the North-South and East-West directions)

We can specify a coordinate plane where North is positive and West is positive (making Soth and East negative).

For F₁, the North-South component is 14sin(66) = 12.7896

For F₁, the East-West component is -14cos(66) = -5.6943

For F₂, the North-South component is -29sin(19) = -9.4415

For F₂, the East-West component is 29cos(19) = 27.4200

Then we can add F₁ North-South with F₂ North-South (because they are in the same direction)

F_{Resultant N-S} = 12.7896 - 9.4415 = 3.3482

And we can add F₁ East-West with F₂ East-West (because they are in the same direction)

F_{Resultant E-W} = -5.6943 + 27.4200 = 21.7257

Then we can put the resultant vector's components back together. The magnitude comes from the Pythagorean Theorem:

F_Resultant = √(3.3482² + 21.7257²) = √483.2173 = 22 N

The direction comes from tan^-1(3.3482/21.7257) = 8.8° above West