Physics Asked by Elvis on November 29, 2020

so I was going over my notes on classical mechanics and just started to review rotation matrices which is the first topic the book starts with. On page 3

The rotation matrix associated with 1.2a and 1.2b is

begin{pmatrix}

costheta & sintheta \

-sintheta & costheta \

end{pmatrix}

but when I try to derive the matrix by following the unit vectors $hat i$ and $hat j$

I get

begin{pmatrix}

costheta & -sintheta \

sintheta & costheta \

end{pmatrix}

The one that the book derives would be clockwise rotation, and the one I got would be for counter-clockwise rotation correct?

You can construct the rotation matrix by finding the direction cosines defined as $lambda_{i,j}=cos(x'_{i},x_{j})$ so:

$lambda_{11}=cos(x'_{1},x_{1})=costheta$

$lambda_{12}=cos(x'_{1},x_{2})=cos(frac{pi}{2}-theta)=sintheta$

$lambda_{21}=cos(x'_{2},x_{1})=cos(frac{pi}{2}+theta)=-sintheta$

$lambda_{22}=cos(x'_{2},x_{2})=costheta$

The book is right.

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