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Transforming Between Unit Coordinate Systems

We can write transformations between unit coordinate systems with coincident origins in terms of direction cosines between axes. If the coordinate system  
\[\mathbf{i} \mathbf{j}\]
  is transformed onto the coorinate system  
\[\mathbf{i'} \mathbf{j'}\]
  then
\[\mathbf{i'} = \mathbf{i} \cdot \mathbf{i'} \mathbf{i} + \mathbf{j} \cdot \mathbf{i'} \mathbf{j}\]

\[\mathbf{j'} = \mathbf{i} \cdot \mathbf{j'} \mathbf{i} + \mathbf{j} \cdot \mathbf{j'} \mathbf{j}\]

The direction cosines are
\[\cos \alpha_{ii'} =\mathbf{i} \cdot \mathbf{i'}\]

\[\cos \alpha_{ij'} =\mathbf{i} \cdot \mathbf{j'}\]

\[\cos \alpha_{ji'} =\mathbf{j} \cdot \mathbf{i'}\]

\[\cos \alpha_{jj'} =\mathbf{j} \cdot \mathbf{j'}\]

We can write the transformation as
\[ \left( \begin{array}{c} \mathbf{i'} \\ \mathbf{j'} \end{array} \right) = \left( \begin{array}{cc} \cos \alpha_{ii'} & \cos \alpha_{ij'} \\ \cos \alpha_{ji'} & \cos \alpha_{jj'} \end{array} \right) \left( \begin{array}{c} \mathbf{i} \\ \mathbf{j} \end{array} \right)\]

Similarly in three dimensions
\[ \left( \begin{array}{c} \mathbf{i'} \\ \mathbf{j'} \\ \mathbf{k'} \end{array} \right) = \left( \begin{array}{ccc} \cos \alpha_{ii'} & \cos \alpha_{ij'} & \cos \alpha_{ik'} \\ \cos \alpha_{ji'} & \cos \alpha_{jj'} & \cos \alpha_{jk'} \\ \cos \alpha_{ki'} & \cos \alpha_{kj'} & \cos \alpha_{kk'}\end{array} \right) \left( \begin{array}{c} \mathbf{i} \\ \mathbf{j} \\ \mathbf{k} \end{array} \right)\]