In physical space, vectors are either properties of an object or actions on an object. Vectors have a magnitude which we represent by a gauge reading and a direction which we represent by an arrow. In vector space we can create geometric representations of vectors which we can then manipulate geometrically. This allows us to geometrically represent any mathematical operations involving vectors, such as vector addition of forces.
The representation of a vector in vector space is similar to the graphical representation of a vector in general except that instead of a just a gauge reading representing the magnitude of the vector, the magnitude is also represented by the length of the arrow. The length of the arrow is based on an arbitrary scale applied consistently to all vectors.
The means of representing vector addition geometrically is called the graphical head-to-tail method and is based on the sides of a triangle. We start with vectors describing a single point or acting on a single point. We redraw these vectors in vector space. Then, for vector addition, we move the vectors such that they form a chain of vectors where the tail of the arrow representing one vector is placed at the head of the arrow representing the other vector, connecting them "head-to-tail" and thus providing two sides of a triangle. The reason we can do this is that parallel vectors with the same magnitude are the same vector, and so if we constrain ourselves to not changing the orientation of the vectors while we manipulate them, then we are geometrically conserving our vectors. This is known as parallel transport and is valid for physical space as well as vector space. The result of the addition operation, or resultant, is the third side of the triangle and points from the tail of the first vector in the chain to the head of the second vector in the chain.
This operation can be continued indefinitely with more vectors added to the chain.