# Matrix Transformations

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Can someone help me with 9 iii please. I’ve worked out the enlargement SF as root2 but not sure how to work out the rotation thanks.

https://imgur.com/a/uUoOMbk

https://imgur.com/a/uUoOMbk

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#2

(Original post by

Can someone help me with 9 iii please. I’ve worked out the enlargement SF as root2 but not sure how to work out the rotation thanks.

https://imgur.com/a/

**Y12_FurtherMaths**)Can someone help me with 9 iii please. I’ve worked out the enlargement SF as root2 but not sure how to work out the rotation thanks.

https://imgur.com/a/

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(Original post by

Do you know the general form of a rotation matrix?

**Notnek**)Do you know the general form of a rotation matrix?

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#4

(Original post by

Yes.

**Y12_FurtherMaths**)Yes.

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So now you know the enlargement matrix (based on the scale factor) and you have the rotation matrix, you can multiply them and equate them to the given matrix. Post your working if you get stuck.

**Notnek**)So now you know the enlargement matrix (based on the scale factor) and you have the rotation matrix, you can multiply them and equate them to the given matrix. Post your working if you get stuck.

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#6

(Original post by

Can someone help me with 9 iii please. I’ve worked out the enlargement SF as root2 but not sure how to work out the rotation thanks.

https://imgur.com/a/uUoOMbk

**Y12_FurtherMaths**)Can someone help me with 9 iii please. I’ve worked out the enlargement SF as root2 but not sure how to work out the rotation thanks.

https://imgur.com/a/uUoOMbk

You can either use the general rotation matrix (which is probably in your formula booklet), or you can draw a small diagram to see that the horizontal basis vector gets mapped to . Draw it on, and determine what angle this vector makes with . (It should be obvious...)

Last edited by RDKGames; 2 years ago

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#7

(Original post by

Ok cheers I’ll try that

**Y12_FurtherMaths**)Ok cheers I’ll try that

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#8

(Original post by

You can just factor out and you will be left with a matrix of rotation.

You can either use the general rotation matrix (which is probably in your formula booklet), or you can draw a small diagram to see that the horizontal basis vector gets mapped to . Draw it on, and determine what angle this vector makes with . (It should be obvious...)

**RDKGames**)You can just factor out and you will be left with a matrix of rotation.

You can either use the general rotation matrix (which is probably in your formula booklet), or you can draw a small diagram to see that the horizontal basis vector gets mapped to . Draw it on, and determine what angle this vector makes with . (It should be obvious...)

EDIT: Actually there's not really any difference

Last edited by Notnek; 2 years ago

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(Original post by

Alternatively you can just sketch how the unit vectors are changed by this matrix and that will give you the angle of rotation quickly.

**Notnek**)Alternatively you can just sketch how the unit vectors are changed by this matrix and that will give you the angle of rotation quickly.

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#10

Just nicer numbers with the original, I guess.

Last edited by RDKGames; 2 years ago

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#11

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How would I do that? I used your method and got 45 degrees anti-clockwise which I’m pretty sure is correct so thanks for that.

**Y12_FurtherMaths**)How would I do that? I used your method and got 45 degrees anti-clockwise which I’m pretty sure is correct so thanks for that.

You should know that the columns of a 2x2 matrix are the images of the unit vectors under the matrix so the unit vector has become and the unit vector becomes .

If you draw a quick sketch showing how these unit vectors change, you'll instantly see the angle of rotation. Also, it's not too hard to see the scale factor from the diagram.

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(Original post by

Yes that's right.

You should know that the columns of a 2x2 matrix are the images of the unit vectors under the matrix so the unit vector has become and the unit vector becomes .

If you draw a quick sketch showing how these unit vectors change, you'll instantly see the angle of rotation. Also, it's not too hard to see the scale factor from the diagram.

**Notnek**)Yes that's right.

You should know that the columns of a 2x2 matrix are the images of the unit vectors under the matrix so the unit vector has become and the unit vector becomes .

If you draw a quick sketch showing how these unit vectors change, you'll instantly see the angle of rotation. Also, it's not too hard to see the scale factor from the diagram.

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#13

(Original post by

Ahh I see how that works with the angle of rotation thanks. How can you tell the scale factor though? (from either diagram)? Without working out the lengths of each line

**Y12_FurtherMaths**)Ahh I see how that works with the angle of rotation thanks. How can you tell the scale factor though? (from either diagram)? Without working out the lengths of each line

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#14

**Y12_FurtherMaths**)

Ahh I see how that works with the angle of rotation thanks. How can you tell the scale factor though? (from either diagram)? Without working out the lengths of each line

(Original post by

If you were to rotate 45 degrees it would still have length 1. But has length so the SF must be .

**Notnek**)If you were to rotate 45 degrees it would still have length 1. But has length so the SF must be .

Inspection can rarely give away the exact values, and this is just one of the nice examples.

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**Notnek**)

If you were to rotate 45 degrees it would still have length 1. But has length so the SF must be .

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**RDKGames**)

You can just factor out and you will be left with a matrix of rotation.

You can either use the general rotation matrix (which is probably in your formula booklet), or you can draw a small diagram to see that the horizontal basis vector gets mapped to . Draw it on, and determine what angle this vector makes with . (It should be obvious...)

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