The graphical method is generally done in one of two ways. Both involve drawing our vectors on a piece of paper. The first way is the tip-to-tail method. We start off by drawing the first vector as an arrow starting at the origin and pointing in the direction of the vector, and with a length of the magnitude of the vector. We then draw the second vector as an arrow, starting at the tip of the first vector, in the direction of the second vector, with a length equal to the magnitude of the second vector.

Then, we can draw a vector from the base of the first vector to the tip
of the second. The direction and magnitude of this final vector will
be the sum of the first two vectors.

An alternative technique is the parallelogram method, in which you
draw both vectors you wish to add starting from the origin, and use them asotwo sides of a parallelogram. The sum of the two vectors is the diagonal of
the parallelogram.

Component Method

While the graphical methods are easy to understand, there is a lot of room for error. Not many people would trust math that required drawing and measuring lines on a piece of paper! A more rigorous way of adding vectors is the component method.

The component method involves taking vectors, breaking them up into x and y coordinates, and adding the x and y coordinates seperately.

For instance, suppose you are giving directions in a downtown area. You leave the office, travel 2 blocks north and 1 block west, to a deli for lunch. After lunch, you walk 1 block south, and 2 blocks west to pick up some computer equipment at the repair shop. When you get there, you realize you need some help carrying it back to the office, so you call an intern at the office to come and help. What directions do you give the intern?

In the component method, this is a simple problem. Just treat north as positive y, south as negative y, east as positive x, and west as negative x. Your initial displacement was x=-1, y=2. You then went x=-2 and y=-1 blocks further. What total displacement from your office did you travel? Just add the x components and y components separately.