CHAPTER 9 - Rectangular Resolution

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.

RECTANGULAR RESOLUTION

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.