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.
Answered by Alberto Navarro on November 29, 2020
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