 # If I am required to compute the full transformation matrix compromising of the following sequence of Quintin Stafford 2022-06-26 Answered
If I am required to compute the full transformation matrix compromising of the following sequence of operations:
rotation by 30 degrees about x-axis
translation by 1, -1, 4 in x, y and z, respectively
rotation by 45 degrees about y axis
Can I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?
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Since this question has been lingering for a long time, I decided to create an answer. I had to make some assumptions since you were not completely specific as to the directions of your rotations. In your case we have (using affine $4×4$ matrices acting on the left of column vectors)
$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \sqrt{3}/2& -1/2& 0\\ 0& 1/2& \sqrt{3}/2& 0\\ 0& 0& 0& 1\end{array}\right]$ (counterclockwise 30 degrees around x)
$\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& -1\\ 0& 0& 1& 4\\ 0& 0& 0& 1\end{array}\right]$ (add (1,-1,4) to the x,y,z components)
$\left[\begin{array}{cccc}\sqrt{2}/2& 0& -\sqrt{2}/2& 0\\ 0& 1& 0& 0\\ \sqrt{2}/2& 0& \sqrt{2}/2& 0\\ 0& 0& 0& 1\end{array}\right]$ (counterclockwise rotation around y)
Multiplying them in the order they are applied, (if the matrices above are A,B,C, then that order is CBA in my notation) that makes a final transformation of
$\left[\begin{array}{cccc}\sqrt{2}/2& -\sqrt{2}/4& -\sqrt{6}/4& -3\sqrt{2}/2\\ 0& \sqrt{3}/2& -1/2& -1\\ \sqrt{2}/2& \sqrt{2}/4& \sqrt{6}/4& 5\sqrt{2}/2\\ 0& 0& 0& 1\end{array}\right]$
With the same matrices, one can see that changing the order of multiplication changes this final transformation.

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