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The original vector u can now be written as






The scalar multipliers u 1, u 2, and u 3 are known as the components of u in the base described by the base vectors e1, e2, and e3. If the base vectors are unit vectors, then the components represent the lengths, respectively, of the three vectors u1, u2, and u3. If the base vectors are unit vectors and are mutually orthogonal, then the base is known as an orthonormal, Euclidean, or Cartesian base.

A vector can be resolved along any two directions in a plane containing it. The figure shows how the parallelogram rule is used to construct vectors a and b that add up to c.

In three dimensions, a vector can be resolved along any three non-coplanar lines. The figure shows how a vector can be resolved along the three directions by first finding a vector in the plane of two of the directions and then resolving this new vector along the two directions in the plane.

When vectors are represented in terms of base vectors and components, addition of two vectors results in the addition of the components of the vectors. Therefore, if the two vectors A and B are represented by

Then,

Rectangular components in 2-D:

The base vectors of a rectangular x-y coordinate system are given by the unit vectors and along the x and y directions, respectively.


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