Possible error in Marion and Thornton's Classical Dynamics?

[Elvis 123456789]

Homework Statement

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, I've uploaded the page here

The rotation matrix associated with 1.2a and 1.2b is

\begin{pmatrix}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta \\
\end{pmatrix}but when I try to derive the matrix by following the unit vectors $\hat i$ and $\hat j$

I get

\begin{pmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta \\
\end{pmatrix}

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

 

[BvU]

No. You want to express the new coordinates in terms of the old with the rotation matrix. The columns are the images of the new unit vectors, expressed in the old coordinate system: $$\hat i' = \cos\theta\; \hat i + \sin\theta\;\hat j \\\hat j' = -\sin\theta\; \hat i + \cos\theta\;\hat j $$