To solve any numerical problems involving vectors, we need to be able to express vectors numerical. The difficulty lies in numerically expressing a physical direction. The solution to this is to set a single direction as a reference and describe all other vectors in relation to that direction.
We start with a single unit vector that denotes the direction "right".
The symbol 
algebraically denotes the "right" unit vector in all of our equations. Unit vectors, by definition, have a magnitude of 1 (see unit circle). Thus any vector that is pointing in the same direction of can be expressed in terms of . More formally, the vectors are said to be parallel.
As can be seen in the diagram, the unit vector 
must be equal to if they are parallel. Generalizing this to all vectors, any vectors with the same magnitude and direction are equal to each other and are, in fact, the same vector.
The next step is to somehow denote the direction "left", but instead of choosing another reference unit vector we use and define any vectors pointing in the opposite direction as being anti-parallel. Algebraically, we denote anti-parallel as being negative. We must note with great importance that the negative sign describes the direction and does not imply a negative magnitude.
We can similarly define the directions "up" and "down" with 
.
Now that we have defined unit vectors, we can numerically, and algebraically, add two or more vectors together by collecting like terms. The easiest vector addition to understand is the addition of force vectors. Simply, if we have two people pulling on an object, an equivalent situation would be a single person pulling on the object where the force is the vector sum of the previous two forces.
