Friday, 8 August 2014

CHAPTER 9 - Rectangular Resolution

Consider two non-collinear vectors \vec a and \vec b; as discussed earlier, these will form a basis of the plane in which they lie. Any vector \vec r in the plane of \vec a and \vec bcan be expressed as a linear combination of \vec a and \vec b
The vectors \overrightarrow {OA}  and \overrightarrow {OB}  are called the components of the vector \vec r along the basis formed by \vec a and \vec b. This is also stated by saying that the vector \vec r when resolved along the basis formed by \vec a and \vec b, gives the components \overrightarrow {OA}  and \,\overrightarrow {OB} . 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 – 20 for a visual picture.


Let us select as the basis for a plane, a pair of unit vector \hat i and \hat j perpendicular to each other.
Any vector \vec r in this basis can be written as
\vec r = \overrightarrow {OA}  + \overrightarrow {OB}
 = \left( {\left| {\vec r} \right|\cos \theta } \right)\hat i + \left( {\left| {\vec r} \right|\sin \theta } \right)\hat j
 = x\hat i + y\hat j
where x and y are referred to as the x and y components of \vec r.
For 3-D  space, we select three unit vectors \hat i,\hat j and \hat k each perpendicular to the other two.
In this case, any vector \vec r will have three corresponding components, generally denoted by xy and z. We thus have
\vec r = x\hat i + y\hat j + z\hat k
The basis (\hat i,\hat j) for the two dimensional case and \left( {\hat i,\hat j,\,\,\hat k} \right) 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.
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