When we mentioned in the introduction that a vector
is either an ordered pair or a triplet of numbers we implicitly defined vectors
in terms of components.
Each entry in the 2-dimensional ordered pair (a, b) or 3-dimensional triplet
(a, b, c) is called a component of the vector. Unless otherwise specified, it is
normally understood that the entries correspond to the number of units the
vector has in the x, y, and (for the 3D case) z directions of a plane or space.
In other words, you can think of the components as simply the coordinates of the
point associated with the vector. (In some sense, the vector is the point, although
when we draw vectors we normally draw an arrow from the origin to the point.)
Vector Addition Using Components
Given two vectors u = (u_{1}, u_{2}) and v = (v_{1}, v_{2}) in the
Euclidean plane, the sum is given by:
u + v = (u_{1} + v_{1}, u_{2} + v_{2})
For three-dimensional vectors u = (u_{1}, u_{2}, u_{3}) and v = (v_{1}, v_{2}, v_{3}), the
formula is almost identical:
u + v = (u_{1} + v_{1}, u_{2} + v_{2}, u_{3} + v_{3})
In other words, vector addition is just like ordinary addition: component by
component.
Notice that if you add together two 2-dimensional vectors you must get another
2-dimensional vector as your answer. Addition of 3-dimensional vectors will
yield 3-dimensional answers. 2- and 3-dimensional vectors belong to different
vector spaces and cannot be added. These same rules apply when we are
dealing with scalar multiplication.