Vector Addition: The Component Method for Vector Addition and Scalar Multiplication

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.

Vector Addition Using Components

Given two vectors u = (u₁, u₂) and v = (v₁, v₂) in the Euclidean plane, the sum is given by:

u + v = (u₁ + v₁, u₂ + v₂)

For three-dimensional vectors u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃), the formula is almost identical:

u + v = (u₁ + v₁, u₂ + v₂, u₃ + v₃)

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.