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I found this comic:

XKCD comic showing a 90°-rotation matrix applied to a column vector equals that column vector drawn sideways

But I can't understand the humor because I can't understand how trig functions affect matrix multiplication. Can someone please explain?

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4  
(+1) for making me laugh... apparently by accident! –  The Chaz 2.0 Jun 17 '11 at 3:23

4 Answers 4

The matrix $$\left[\begin{align} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{align} \right]$$ when it acts on a vector it rotates the vector by $\theta$ in clockwise direction. Hence when $\theta = 90^{\circ}$, it rotates the vector $$\left[\begin{align} a_1 \\ a_2 \end{align} \right]$$ from vertical to horizontal clockwise.

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$\theta=-90$... –  lhf Jun 3 '11 at 19:12
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@lhf: Thats why I have written clockwise direction. –  user17762 Jun 3 '11 at 19:14
    
Oops, missed that clockwise. –  lhf Jun 3 '11 at 19:20

Well, the rotation matrix they write there is

$$ \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} \right) $$

If you multiply a vector $\left( \begin{array}{c} a_{1} \\ a_{2} \\ \end{array} \right)$ by this matrix then you end up with $\left( \begin{array}{c} a_{2} \\ -a_{1} \\ \end{array} \right)$ If you draw a picture in the $xy$ plane connecting each of $(a_{1},a_{2})$ and $(a_{2},-a_{1})$ with the origin, it will be clear that the latter is a 90 degree rotation of the former.

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That's a $-90$-degree rotation matrix.

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Wow! A highly improbable case of two events occurring simultaneously! Your answer and mine show the same "answered 2 secs ago"! –  user17762 Jun 3 '11 at 19:11

Because...

enter image description here

Sorry couldn't resist when it was bumped.

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