the following force vectors act on an object?

Question

i) 50.0 newtons at 45 degrees north of east and ii) 25.0 newtons at 30 degrees south of east. what is the magnitude and angle of resultant relative to the easterly direction?

Answer 1

Method:
1. Resolve each vector into its North-South and East-West components.
2. Add the N-S components, and add the E-W components, so that there is only one N-S and one E-W component.
3. Combine the single N-S vector and the single E-W vector to determine the magnitude and direction of the resultant vector.

Application of method to the values given:
1. 50 N at 45 degrees E of N equals an N component equal to 50sqrt(2) (where square root of 2 is sin 45 degrees), and an E component equal to 50sqrt(2) (where this square root of 2 equals cos 45 degrees).
25 N at 30 deg S of E equals an S component equal to 25 x .5 (where .5 = sin 30 deg), and a W component equal to 25sqrt(3)/2 (where (square root of 3)/2 equals cos 30 deg).
2. Combine the N and S components. (N will be larger, so subtract the S component from the N component to get the net component in a northerly direction.)
Combine the E and W components. (E will be larger, so subtract the W component from the E component to get the net component in an easterly direction.)
3. The magnitude of the resultant is found by the Pythagorean theorem. (Square the N and E components. Add the squares. Take the square root of the sum. That is the magnitude of the resultant force.) Call this answer F (the force is F newtons).
The direction of the resultant can be found by trigonometry. The N component divided by the E component is the tangent of the angle between the resultant force and due east. So divide N by E and take that arc sin. Call this angle A.
Then the answer is "a force of F newtons in a direction A degrees north of east."

Answer 2

Magnitude of the Resultant force:
Rx = F1x + F2x = 50*cos 45 +25 cos 30
= 50*0.7071 +25*0.8660 =35.355 +21.65=57.005 N
Ry = F1Y - F2y =50*sin 45 - 25 sin 30 =50*0.7071 - 12.5
= 35.355 -12.5 = 22. 855 N
R = sqrt sum of the squares of Rx and Ry = 61.4 N
Direction: Angle A with the East;
Tan A =Ry/Rx = 22.855/57.005 = 0.4009
Angle A = tan -1(0.4009) = 21.85 degrees North of East.