How to Add Vectors?

We know vectors differ from scalar because it not only has a magnitude, but also a direction. On a graph, we will assume that the positive directions for the y axis is north (up), and the positive direction for the x axis is east (right).Therefore, directions of south and west symbolize negative vectors.
Now for adding vectors on a graph. In most cases, there were be more than 1 or 2 vectors heading into all directions. If you are a great mathematician like Mr. Dulmage, you would say, "Oh, I can solve this trignonmetry, EASILY, because the you first get the sine of blah blah, and use that to get the cosine of blah blah..."
Confusing isn't it? Most of us aren't great mathematicians, or are at some extent.Therefore, a much simpler to calculate the resultant is to simply "add" all the x values and y values of vectors. (Note: in some cases when vector is negative, we're theoritically adding vectors, but adding by a negative digit is also the same as subtracting the number).
So next we'll have to figure out how to find the x and y values for the vectors on the graph to add.
In one case, if the vector's x and y values are given, that's great because you wouldn't need to worry about calculating them. But when they're not given, we'll use some trignometry we learned in grade 10 math to solve this problem.

Opposite side is the y value, adjacent side is the x value. With basic sin and cos laws, we would be able to calculate the x and y values.
So next as mentioned already, we add all the x and y values. But keep in mind that even when some times you're adding a vector, you have to be careful because the direction of the vector determine if the number is positive or negative.

 For the x axis, vectors with direction east are positive. Vice versa, vectors with direction west are negative. When considering only the x value for adding two vectors, if both vectors have direction east, you add the 2 numbers, if both have direction east, it would be (-x1)+(-x2), so theoritically adding 2 negative numbers. When one is east and one west, it will be one positive subtract one negative.
As demonstrated by the diagram, adding y values for the resultant y would be the same way.

Finall, now after all that, we get our resultant x and y values. But we're not done yet because we have to calculate the hypotenuse. We can do this with the help of Pythagora and his theorem.

 

So now we are able to calculate the hypotenuse with the derived formula x2 + y2 = R2, but we still have to know which direction specifically the vector is heading towards. This can be determined by going back to the x and y values for the resultant.Once again, positive y value means the vector is heading towards north, and vice versa. Positve x value means the vector is heading towards east, and vice versa.
So after we get these 2 directions, we can imagine a little triangle inside our head: If the vector start from the origin (0,0), by what degree have it gone east or west from south or north?

This can be again calculated from the help of trignometry. We know that the tan is calculated from opposite over adjacent, so now we just have to see the triangle in different views to see which value (x or y) is the opposite side and adjacent side. For this calculation, make sure that you're reading the angle from the y axis to the x axis with respect to where the vector is located.
And now finally, we write the resultant vector as: the hypotenuse value [ydirection (degree) xdirection]

If you're still confused because I know I didn't explain very well, check out this video - it MIGHT help you :)
http://www.youtube.com/watch?v=UmLu3HIKBG8