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Cartesian unit vectors expressed by Cylindrical unit vectors

  1. Oct 1, 2014 #1
    please someone explain me the following expression for Cartesian unit vectors expressed by the cylindrical unit vectors:

    http://web.mit.edu/8.02t/www/materials/modules/ReviewB.pdf
    at page B-8 line B.2.4

    i would like to know which steps led to it.

    thanks,

    Chen
     
  2. jcsd
  3. Oct 1, 2014 #2

    Mark44

    Staff: Mentor

    One way to think of this is in terms of matrices:
    $$ \begin{bmatrix} \hat{\rho} \\ \hat{\phi} \end{bmatrix} = \begin{bmatrix} cos(\phi) & sin(\phi) \\ -sin(\phi) & cos(\phi) \end{bmatrix} \begin{bmatrix} \hat{i} \\ \hat{j}\end{bmatrix}$$
    Apply the inverse of the above matrix to get a vector with the unit vectors i and j. The inverse is:
    $$\begin{bmatrix} cos(\phi) & -sin(\phi) \\ -sin(\phi) & cos(\phi) \end{bmatrix} $$
     
  4. Oct 1, 2014 #3
    I tried to understand it by the geometry of it
    but that's a better way to understand it.

    Thanks
     
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