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matrices question

ok this seems like bit of a stupid question cuz i feel like the answer is obvious, but i was wondering if there's a way to distinguish between rotation and reflection matrices. cuz i hv a test tomoz, and i feel like im gonna get confused between whether something is a rotation or a reflection? like i know you the columns are the images of the identity matrix, but how do i know when to use the t=roation matrix? (cos theta,-sin theta etc...)

many thanks
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Avatar for mqb2766
mqb2766
21
Original post by vix.xvi
ok this seems like bit of a stupid question cuz i feel like the answer is obvious, but i was wondering if there's a way to distinguish between rotation and reflection matrices. cuz i hv a test tomoz, and i feel like im gonna get confused between whether something is a rotation or a reflection? like i know you the columns are the images of the identity matrix, but how do i know when to use the t=roation matrix? (cos theta,-sin theta etc...)

many thanks


The rotation/reflection matrix has a huge amount of symmetry as there is only 1 effective parameter (theta). You should be able to easily check?
https://en.m.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions
Rotation has equal diagonal elements and opposing signs for the other diagonal.
Reflection is the opposite.
(edited 3 years ago)
Original post by vix.xvi
ok this seems like bit of a stupid question cuz i feel like the answer is obvious, but i was wondering if there's a way to distinguish between rotation and reflection matrices. cuz i hv a test tomoz, and i feel like im gonna get confused between whether something is a rotation or a reflection? like i know you the columns are the images of the identity matrix, but how do i know when to use the t=roation matrix? (cos theta,-sin theta etc...)

many thanks

Find the determinant. If negative, a reflection is involved.
Original post by RDKGames
Find the determinant. If negative, a reflection is involved.

thank you!!
Original post by mqb2766
The rotation/reflection matrix has a huge amount of symmetry as there is only 1 effective parameter (theta). You should be able to easily check?
https://en.m.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions
Rotation has equal diagonal elements and opposing signs for the other diagonal.
Reflection is the opposite.
Original post by RDKGames
Find the determinant. If negative, a reflection is involved.


Original post by mqb2766
The rotation/reflection matrix has a huge amount of symmetry as there is only 1 effective parameter (theta). You should be able to easily check?
https://en.m.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions
Rotation has equal diagonal elements and opposing signs for the other diagonal.
Reflection is the opposite.

Sorry to disturb again, but i had another q, hope you dont mind :smile:

for a matrix like this (see attached), how do i know if i should draw out the points 1,00 , 0,1,1 , etc... or if i should try the rotation matrix?

also the mark scheme says that that matrix is reflection in the line y=-x, but i also got that its a rotation 270* Clockwise about Z, who is correct?

thanks :h:
WhatsApp Image 2020-11-08 at 11.51.53 AM.jpeg92.1KB
Mark scheme is correct.

Edit: the behaviour of the matrix is completely defined by its effect on the vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1), so there's nothing wrong with looking at these as long as you are able to correctly work out what geometric transformation is implied by what happens to those vectors. If you're getting different answers, it's because you're doing the bit in bold wrong. (Sometimes it will be difficult-to-impossible to work out the geometric transformation. That's when you need to use more advanced mathematics. But I'm not really sure what's covered in the A-level syllabus).
(edited 3 years ago)
Reply 6
Avatar for mqb2766
mqb2766
21
Original post by vix.xvi
Sorry to disturb again, but i had another q, hope you dont mind :smile:

for a matrix like this (see attached), how do i know if i should draw out the points 1,00 , 0,1,1 , etc... or if i should try the rotation matrix?

also the mark scheme says that that matrix is reflection in the line y=-x, but i also got that its a rotation 270* Clockwise about Z, who is correct?

thanks :h:

As DFranklin said ...
Also the 2*2 x-y submatrix is not a rotation about z. The diagonal elements are equal (0) as are the other diagonal elements (-1). A rotation would have +/-1 for the other diagonal.
See the examples in the previous wiki link.

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