Rotation matrix

**noles2188** · September 15th 2009, 08:36 PM

Find the rotation matrix that C that transforms the matrix
|0| into |3|
|5| |4|

(the first matrix is 0 5 and the second has 3 4, each vertical)

**Bruno J.** · September 15th 2009, 08:53 PM

Hint : look at the vectors (0,5) and (3,4) in the (x,y) plane. What is the angle between them? That's the angle you're looking for; then just find the corresponding rotation matrix.

**HallsofIvy** · September 16th 2009, 06:15 AM

Another way to do this: Any "rotation" matrix in the plane is of the form
$\begin{bmatrix}a & -b \\ b & a\end{bmatrix}$
with $a^2+ b^2= 1$ .
$\begin{bmatrix}a & -b \\ b & a\end{bmatrix}\begin{bmatrix}0 \\ 5\end{bmatrix}=\begin{bmatrix}-5b \\ 5a\end{bmatrix}= \begin{bmatrix}3 \\ 4\end{bmatrix}$
gives you two equations to solve for a and b. Since the lengths of the two vectors, <0, 5> and <3, 5> are the same, $a^2+ b^2= 1$ will automatically be satisfied.

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