# Linear Transformations Edexcel

This might be a stupid question, but on the formula booklet, theres a formula under matrix transformations:

Reflection in the line y = tan(theta)x :
(cos(2theta) sin(2theta))
(sin(2theta) -cos(2theta))

but this isn't mentioned anywhere in the textbook as far as I'm aware

What's the formula for and where does it come up because I've just looked through all of the matrix and linear transformations stuff and it's not mentioned anywhere
Original post by Amy.fallowfield
This might be a stupid question, but on the formula booklet, theres a formula under matrix transformations:

Reflection in the line y = tan(theta)x :
(cos(2theta) sin(2theta))
(sin(2theta) -cos(2theta))

but this isn't mentioned anywhere in the textbook as far as I'm aware

What's the formula for and where does it come up because I've just looked through all of the matrix and linear transformations stuff and it's not mentioned anywhere

Further Maths content.

See page 12 on https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-further-mathematics-specification.pdf

"For 2-D, identification and use of the matrix representation of single and combined transformations from: reflection in coordinate axes and lines y = ± x, ..."

That formula you see combines all of these reflection lines into one expression.

For refl in x-axis you set theta=0 and use the matrix you get.

For refl in y=x you set theta=45 and use the matrix you get.

For refl in y-axis you set theta=90 and use the matrix you get.

Etc...

Key to use it is to treat theta as the anticlockwise angle from the positive x-axis to your line of reflection. Once you know what it is, your reflection matrix pops out.
(edited 6 months ago)
Original post by RDKGames
Further Maths content.

See page 12 on https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-further-mathematics-specification.pdf

"For 2-D, identification and use of the matrix representation of single and combined transformations from: reflection in coordinate axes and lines y = ± x, ..."

That formula you see combines all of these reflection lines into one expression.

For refl in x-axis you set theta=0 and use the matrix you get.

For refl in y=x you set theta=45 and use the matrix you get.

For refl in y-axis you set theta=90 and use the matrix you get.

Etc...

Key to use it is to treat theta as the anticlockwise angle from the positive x-axis to your line of reflection. Once you know what it is, your reflection matrix pops out.

Oh I get it. Thank you!
You can always double-check by multiplying the transformation matrix with the basis vectors. Then, see if it matches the geometric meaning (preferably through drawing a picture). In this general case though, since you are not reflecting along "a nice line", angle chasing could be a bit tricky.

See my other comment if you want an example.
(edited 6 months ago)
Original post by tonyiptony
You can always double-check by multiplying the transformation matrix with the basis vectors. Then, see if it matches the geometric meaning (preferably through drawing a picture). In this general case though, since you are not reflecting along "a nice line", angle chasing could be a bit tricky.

See my other comment if you want an example.

Just for completeness, its reasonably well described in
https://en.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions
and you can view a reflection
ref(theta)
as a ref(0) (reflection in the x axis) followed by a rot(2theta) as should be reasonably clear from the original matrix / thinking about the basis vectors so ref(0) followed by a rot(2theta) gives (vectors are columns)
(1,0) -> (1,0) -> (cos(2theta),sin(2theta))
(0,1) -> (0,-1) -> (sin(2theta),-cos(2theta))
So the original reflection matrix.

Alternatively, by thinking about the (1,0) and (0,1) seperately, a sketch should show (1,0) would be rotated by 2theta and (0,1) would be rotated by 2(-pi/2+theta) which gives the same result.
(edited 6 months ago)