Question: Add the vectors using components. 3 N in a direction 15° south of west and 4 N in a direction 12° east of south.

I have attempted the question and got a resultant of (3.136, -2.06) from (3cos255°, 3sin255°)+(-4cos168°,4sin168°). I do not know if I did it correctly. Please provide a step-by-step solution for me to understand how to do these type of questions on my own. Thanks!

Translating your directions into the standard trig notation, where East is 0°

and counterclockwise would be a positive rotation ....

15° south of west --- to me meant: 195°
12° east of south--- to me meant : 282°

so (3cos195,3sin195) + (4cos282,4sin282)
= (-2.066... , -4.689..)

I won't even attempt to figure out where you got your angles of 255 and 168 from.
You must have taken North to be 0° and rotate clockwise?

If you draw the diagram, you can see that your first vector 3@155° puts you in the third quadrant at (-3cos15°,-3sin15°)

Measuring u° clockwise (from due North = 0°), the standard angle of v° (counterclockwise from the +x direction) is v = 90°-u. That would give you
(3cos(90-255)°,3sin(90-255)°) = (3cos(-165°),3sin(-165°)) = (-3cos15°,-3sin15°)

So you have the right idea, but you need to translate between the two orientations.

All angles are measured CCW from +x-axis.

X = 3*Cos195+4*Cos282 = -2.07 N.
Y = 3*sin195+4*sin282 = -4.69 N.
Fr = sqrt((-2.07)^2 + (-4.69)^2 = 5.13 N. = Resultant force.

To add vectors using components, you need to break down each vector into its x and y components, and then add the corresponding components together to get the resultant vector.

Step 1: Convert the magnitudes and directions of each vector into x and y components.
For the first vector (3 N, 15° south of west), the x and y components can be found using trigonometry.

The x component is given by: X₁ = Magnitude₁ * cos(θ)
X₁ = 3 N * cos(255°) = -2.598 N

The y component is found by: Y₁ = Magnitude₁ * sin(θ)
Y₁ = 3 N * sin(255°) = -0.749 N

Similarly, for the second vector (4 N, 12° east of south):

The x component is: X₂ = Magnitude₂ * cos(θ)
X₂ = 4 N * cos(168°) = 3.784 N

The y component is: Y₂ = Magnitude₂ * sin(θ)
Y₂ = 4 N * sin(168°) = -2.06 N

Step 2: Add the x-components together.
X_total = X₁ + X₂
X_total = -2.598 N + 3.784 N = 1.186 N

Step 3: Add the y-components together.
Y_total = Y₁ + Y₂
Y_total = -0.749 N + (-2.06 N) = -2.809 N

Step 4: Combine the x and y components to find the resultant vector.
The resultant vector is given by: (X_total, Y_total)
Res = (1.186 N, -2.809 N)

So, based on these calculations, the resultant vector of adding these two vectors using components is approximately (1.186 N, -2.809 N).

I hope this step-by-step explanation helps you understand how to approach and solve problems like these in the future.