Consider two non-collinear vectors and ; as discussed earlier, these will form a basis of the plane in which they lie. Any vector in the plane of and can be expressed as a linear combination of and
The vectors and are called the components of the vector along the basis formed by and . This is also stated by saying that the vector when resolved along the basis formed by and , gives the components and . Also, as discussed earlier, the resolution of any vector along a given basis will be unique.
We can extend this to the three dimensional case: an arbitrary vector can be resolved along the basis formed by any three non-coplanar vectors, giving us three corresponding components. Refer to Fig – for a visual picture.
Let us select as the basis for a plane, a pair of unit vector and perpendicular to each other.
Any vector in this basis can be written as
where and are referred to as the and components of .
For space, we select three unit vectors and each perpendicular to the other two.
In this case, any vector will have three corresponding components, generally denoted by , and . We thus have
The basis for the two dimensional case and for the three-dimensional case are referred to as rectangular basis and are extremely convenient to work with. Unless otherwise stated, we’ll always be using a rectangular basis from now on. Also, we’ll always be implicitly assuming that we’re working in three dimensions since that automatically covers the two dimensional case.