To calculate the resultant of two vectors A and B, we need to follow these steps:

1. Find the horizontal and vertical components of each vector


2. Add the horizontal components and the vertical components separately


3. Use Pythagoras' theorem to find the magnitude of the resultant vector


4. Use trigonometry to find the angle that the resultant vector makes with the positive x-axis

To find the horizontal and vertical components of vector A, we use the following formulas:

Ax = A cos θ

Ay = A sin θ

where A is the magnitude of vector A and θ is the angle that vector A makes with the positive x-axis.

Substituting the given values, we get:

Ax = 3 cos 20°

Ay = 3 sin 20°

Ax = 2.817 m

Ay = 1.026 m



Using the same formulas as above, we can find the horizontal and vertical components of vector B:

Bx = B cos θ

By = B sin θ

Substituting the given values, we get:

Bx = 4 cos 110°

By = 4 sin 110°

Bx = -0.984 m

By = 3.640 m



To find the horizontal and vertical components of the resultant vector R, we simply add the corresponding components of vectors A and B:

Rx = Ax + Bx

Ry = Ay + By

Substituting the values, we get:

Rx = 2.817 - 0.984

Ry = 1.026 + 3.640

Rx = 1.833 m

Ry = 4.666 m



Using Pythagoras' theorem, we can find the magnitude of the resultant vector R:

R = sqrt(Rx^2 + Ry^2)

Substituting the values, we get:

R = sqrt((1.833)^2 + (4.666)^2)

R = 4.926 m



To find the angle that the resultant vector R makes with the positive x-axis, we use trigonometry:

θ = tan^-1(Ry/Rx)

Substituting the values, we get:

θ = tan^-1(4.666/1.833)

θ = 69.03°

Therefore, the resultant vector R has a magnitude of 4.926 m and makes an angle of 69.03° with the positive x-axis.