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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.)

Figure %: The vector (*a*, *b*) in the Euclidean plane.

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.

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