Plane of reflection.
Hi,
I need to solve the following:
Mr=r and Nr=r where r is the column matrix x,y,z. To find the axis of rotation and plane of reflection respectively. M is a rotation matrix given by:
M=
and N is a reflection matrix given by:
.
I have made a thread on this previously where I did work out the answer to the axis of rotation: http://www.thestudentroom.co.uk/showthread.php?t=2318209&page=3&p=42224810#post42224810
However when I apply a similar method to find the plane of reflection I end up with the matrix:
. Which I'm not too sure how to interpret. I then tried to solve it using eigenvectors, but I think I'm doing something wrong as my answer doesn't come out to be the same as what I got for the axis of rotation in the previous thread which is x=-y=z.
Any help would be appreciated. Thank you!
I need to solve the following:
Mr=r and Nr=r where r is the column matrix x,y,z. To find the axis of rotation and plane of reflection respectively. M is a rotation matrix given by:
M=
and N is a reflection matrix given by:
.
I have made a thread on this previously where I did work out the answer to the axis of rotation: http://www.thestudentroom.co.uk/showthread.php?t=2318209&page=3&p=42224810#post42224810
However when I apply a similar method to find the plane of reflection I end up with the matrix:
. Which I'm not too sure how to interpret. I then tried to solve it using eigenvectors, but I think I'm doing something wrong as my answer doesn't come out to be the same as what I got for the axis of rotation in the previous thread which is x=-y=z.
Any help would be appreciated. Thank you!
Original post by Music99
...
Your first matrix isn't M, but rather M-I, which is what you needed at the time.
Assuming you've gone from N to N-I and solved correctly, you're looking for the equation of a plane this time - it won't be the same answer as before, or even the same format.
What happens when you translate that matrix you've created (setting it = 0) into a cartesian equation?
Done!
Original post by ghostwalker
Your first matrix isn't M, but rather M-I, which is what you needed at the time.
Assuming you've gone from N to N-I and solved correctly, you're looking for the equation of a plane this time - it won't be the same answer as before, or even the same format.
What happens when you translate that matrix you've created (setting it = 0) into a cartesian equation?
Done!
Assuming you've gone from N to N-I and solved correctly, you're looking for the equation of a plane this time - it won't be the same answer as before, or even the same format.
What happens when you translate that matrix you've created (setting it = 0) into a cartesian equation?
Done!
Yeah I did minus I from it, I was just writing what M and N were before subtracting I. So once I subtract the Identity matrix from N, and did the Gauss Jordan elimination I ended up with the 1 -1 1
0 0 0
0 0 0
Which If translated into the equation of a plane is x=-y=z which is the same as the axis of rotation?
(edited 11 years ago)
Original post by Music99
Which If translated into the equation of a plane is x=-y=z which is the same as the axis of rotation?
Which If translated into the equation of a plane is x=-y=z which is the same as the axis of rotation?
Er, no.
What's
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
Original post by ghostwalker
Er, no.
What's
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
What's
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
well if by the T you mean the transpose, then the transpose of that above is
1 0 0
-1 0 0
1 0 0
Original post by Music99
well if by the T you mean the transpose, then the transpose of that above is
1 0 0
-1 0 0
1 0 0
1 0 0
-1 0 0
1 0 0
T is transpose.
I means multiply the two matrices together.
Original post by ghostwalker
T is transpose.
I means multiply the two matrices together.
I means multiply the two matrices together.
Ah okay so when I do the transpose first and multiply it by the matrix I get
1 -1 1
-1 1 -1
1 -1 1
and if I do the matrix first and multiply by the transpose I get
3 0 0
0 0 0
0 0 0
So I'm assuming it's the top one that I want. So then we get:
x=-y=z
-x=y=-z
x=-y=z
Original post by Music99
Ah okay so when I do the transpose first and multiply it by the matrix I get
1 -1 1
-1 1 -1
1 -1 1
and if I do the matrix first and multiply by the transpose I get
3 0 0
0 0 0
0 0 0
So I'm assuming it's the top one that I want. So then we get:
x=-y=z
-x=y=-z
x=-y=z
1 -1 1
-1 1 -1
1 -1 1
and if I do the matrix first and multiply by the transpose I get
3 0 0
0 0 0
0 0 0
So I'm assuming it's the top one that I want. So then we get:
x=-y=z
-x=y=-z
x=-y=z
It's neither. You really do need to look at basic matrix arithmetic.
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
x-y+z for the first co-ordinate, and the other two being 0, and that is equal to 0 from what would be the zero vector on the right of the Ax=0 equation.
So, the equation of your plane is x-y+z=0
(edited 11 years ago)
Original post by ghostwalker
It's neither. You really do need to look at basic matrix arithmetic.
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
x-y+z for the first co-ordinate, and the other two being 0, and that is equal to 0 from what would be the zero vector on the right of the Ax=0 equation.
So, the equation of your plane is x-y+z=0
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
x-y+z for the first co-ordinate, and the other two being 0, and that is equal to 0 from what would be the zero vector on the right of the Ax=0 equation.
So, the equation of your plane is x-y+z=0
Ah I didn't realise it was the x,z,y matrix, misread what you put. I do get that when I multiply out the matrices though. Thank you.
Original post by ghostwalker
It's neither. You really do need to look at basic matrix arithmetic.
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
x-y+z for the first co-ordinate, and the other two being 0, and that is equal to 0 from what would be the zero vector on the right of the Ax=0 equation.
So, the equation of your plane is x-y+z=0
1-1 1
0 0 0
0 0 0
times (x,y,z)^T
x-y+z for the first co-ordinate, and the other two being 0, and that is equal to 0 from what would be the zero vector on the right of the Ax=0 equation.
So, the equation of your plane is x-y+z=0
It's weird how the axis of rotation and plane of reflection come out as the same equation more or less
.Original post by Music99
It's weird how the axis of rotation and plane of reflection come out as the same equation more or less .
.Consider, what's a normal to the plane of reflection?
Original post by ghostwalker
Consider, what's a normal to the plane of reflection?
The origin would work no?
Original post by Music99
The origin would work no?
No.
I think you're missing too much background information, and really need to look up work on the plane in cartesian and vector form.
Original post by ghostwalker
No.
I think you're missing too much background information, and really need to look up work on the plane in cartesian and vector form.
I think you're missing too much background information, and really need to look up work on the plane in cartesian and vector form.
That's the thing, like If im given a question involving a plane in cartesian form or I'm asked to find the normal of the plane I can do it fine. But when it's in matrix form I get all confused.
Original post by Music99
That's the thing, like If im given a question involving a plane in cartesian form or I'm asked to find the normal of the plane I can do it fine. But when it's in matrix form I get all confused.
Erm, well, x-y+z=0 is in cartesian form.
Original post by ghostwalker
Erm, well, x-y+z=0 is in cartesian form.
so 1,-1,1
Original post by Music99
so 1,-1,1
Yep.
Quick Reply
Related discussions
- vectors
- Isaac Physics - Parabolic Mirror
- A-level Biology Question (AQA)
- Design & Technology NEA
- A Level Mechanics
- Further maths vectors. Proving 2 lines lie in the same plane
- Intersection of planes / normals question
- What 3 albums would you take if you were going to be stuck on a desert island?
- Maths - mechanics
- Further Maths Vectors Help
- A-Level Mechanics Help
- Friction Question Help
- Please help with this math question
- The learning effect question help
- Mechanical vs Aerospace
- Help mechanics
- Force Diagram (A2 Edexcel)
- A level physics
- 3D Vectors
- Further math vecotr question
Latest
Trending
Last reply 1 day ago
Did Cambridge maths students find maths and further maths a level very easy?Last reply 2 weeks ago
Edexcel A Level Mathematics Paper 2 unofficial mark scheme correct me if wrongMaths
71
Trending
Last reply 1 day ago
Did Cambridge maths students find maths and further maths a level very easy?Last reply 2 weeks ago
Edexcel A Level Mathematics Paper 2 unofficial mark scheme correct me if wrongMaths
71
