Cartesian unit vectors expressed by Cylindrical unit vectors

[chenrim]

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

 

[Mark44]

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

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:
$$\begin{bmatrix} cos(\phi) & -sin(\phi) \\ -sin(\phi) & cos(\phi) \end{bmatrix} $$

 

[chenrim]

I tried to understand it by the geometry of it
but that's a better way to understand it.

Thanks