Vector sum question

There are two basic ways you can approach this. You can do tip to tail addition, which is arranging one vector's tip (leading end) to another vectors tail, and the resultant will be from the tail of the first to the tip of the latter, and you can solve the resulting triangle using various trig methods.

The other way it to graph these on coordinate planes, and treat each vector as if it were the hypotenuse of a right triangle, where the horizontal leg is the x component of the vector and the vertical leg is the y component. The sum of the x components of the vectors will be the x component of the resultant vector. The sum of the y components will be the y component of the resultant vector, and you can use Pythagorean’s theorem and trig ratios to solve the magnitude and direction of the resultant.

 

Positives and negatives in vectors are directional indicators - nothing else. -5 m /north is the same as 5 m /south. The problem as you've expressed it is to look for the vector sum, so there is no subtraction involved - at least, not directly.

One possible approach not mentioned is to resolve all three vectors into north/south and east/west components, add those, then determine the resultant. In that case, some of the components may well be negative - you'll still add them, but adding a negative is subtraction (she said, trying not to sound too stupid saying it).

Doing the tip-to-tail method will give you a four-sided figure (including the resultant). If you're going to do it that way, I would suggest you pick two of them and add them (you'll have a triangle, and basic trigonometry will help immeasurably. Don't forget the Law of Sines and the Law of Cosines), then add the resultant to the third. Working with four of them at once in polar notation usually results in a quick trip to the nearest asylum for the terminally bewildered. Converting the polar coordinates you have to Cartesian coordinates is functionally equivalent to my first suggestion.

 

"Sum" means add whether you are working with numbers or vectors! What you are really asking is when the components are positive and when they are negative. That's determined when you set up your coordinate system- here you are using the "standard"- East is the positive x direction, North is the positive y direction- and so anything West or South is negative.
Do you have much experience with drawing graphs? On a standard map, East is to the right, West left; North is up, South down.
On a standard graph, positive x is right, negative x left; positive y is up, negative down.

The components of the first vector are (-2.4 cos(32), -2.4 sin(32)): both negative because the vector is "south of west". You had everything right except the sign.
The components of the second vector are (0, -1.6): notice the 0! The second vector is due South (which is why the y-component is negative) so there is no "east" or "west" and no x-component.
The components of the third vector are (4.9 cos(27), -4.9 sin(27)). (Take a look at your triangle. You said the vector was "27 degrees S of E". Using 63 degrees is measuring the angle from the South and then you swap sine and cosine.) Now the x-component is positive since the vector is "E" and the y-component is positive because the vector is "S".

The "sum" is (-2.4 cos(32)+ 0+ 4.9 cos(27), -2.4 sin(32)- 1.6-4.9 sin(27)).

It's not a matter of "adding" or "subtracting", it's a matter of whether the numbers themselves are positive or negative- and that is based on "North positive, South negative; East positive, West negative".