## 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 $\vec b$can 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.

#### RECTANGULAR RESOLUTION

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 $x$ $y$ 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.