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I have a problem understanding this.

Question: Find the matrix $A$ of the linear transformation from $\Bbb R^2 \to \Bbb R^2$ that rotates any vector through an angle of $150^\circ$ in the clockwise direction.

The standard matrix:

$$\begin{pmatrix}\cosθ &\!\!\!-\sinθ\\ \sinθ &\cosθ\end{pmatrix}$$

Plugging in the value $150$ for the angle does not work and I'm not sure why at this point. Any input?

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Hint:

Your matrix rotates a vector in the plane in angle $\,\theta\,$ ...counterclockwise ...!

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  • $\begingroup$ Would that just mean to reverse both sins ? $\endgroup$
    – Dimitri
    Mar 28 '13 at 20:50
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    $\begingroup$ If you meant to ask whether multiplying the sines by $\,-1\,$ will do the job clockwise the answer is yes, but more important imo: do you understand, say geometrically, why ? $\endgroup$
    – DonAntonio
    Mar 28 '13 at 20:52
  • $\begingroup$ Yes, I was actually in the midst of checking the geometry as we speak. $\endgroup$
    – Dimitri
    Mar 28 '13 at 20:57
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The units on the argument for sine and cosine is in radians. You need to convert your 150 degrees to radians if you want to rotate by the proper amount.

$$Radians = \dfrac{\pi}{180}Degrees$$

Furthermore, your matrix is set up to transform in the counterclockwise direction. If you want to rotate in the opposite direction, try plugging in a $-\theta$ and see what that gets you.

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  • $\begingroup$ You're absolutely right about that. Thank you! $\endgroup$
    – Dimitri
    Mar 28 '13 at 20:52

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