Composite Matrix Transformation - Reflection

[lubricarret]

Homework Statement

Let T1 be the reflection about the line 2x–5y=0 and T2 be the reflection about the line –4x+3y=0 in the euclidean plane.
(i) The standard matrix of T1 o T2 is: ?

Thus T1 o T2 is a counterclockwise rotation about the origin by an angle of _ radians?

(ii) The standard matrix of T2 o T1 is: ?

Thus T2 o T1 is a counterclockwise rotation about the origin by an angle of _ radians?

Homework Equations

I think these equations are correct...

T(v) = A(v)

Reflection:
A =
[((2(u_1))^2)), (2(u_1)(u_2)))
(2(u_1)(u_2)), ((2(u_2))^2))]
*u being the unit vectors

Rotation counterclockwise:
A =
[cosx -sinx
sinx cosx]

S o T is the matrix Transformation with matrix AB

The Attempt at a Solution

I thought I understood this, but again, I guess I've understood something incorrectly.

For the first question, I got the unit vectors to be:
[(5/sqrt29)], (2/sqrt29)] and [(3/5), (4/5)] for T_1 and T_2 respectively.
I then got the standard matrix A of T_1 to be:
[(21/29) (20/29)
(20/29) (-21/29)]
and the standard matrix B of T_2 to be:
[(-7/25) (24/25)
(24/25) (7/25)]

I then took AB = the dot product of these matrices to get:
[(333/6350) (644/6350)
(-644/6350) (333/6350)]

I did similar for the second part, but I'll spare all the numbers, since I'm messing something up...

Further, how would I go about getting the radians? I know the formula for counterclockwise rotation, but wouldn't know how to come up with the radians of such a number...

 

[HallsofIvy]

I don't recognize your formula for the reflection matrix so what I would do is this.
<3, 4> is a vector in the direction of the line -4x+ 3y= 0 and <-4, 3> is a vector perpendicular to it. The reflection in that line maps <3, 4> into itself and <-4, 3> into its negative, <4, -3> Setting up the two equations
$$\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\left[\begin{array}{c}3 \\ 4\end{array}\right]= \left[\begin{array}{c}3 \\ 4\end{array}\right]$$
and
$$\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\left[\begin{array}{c}-4 \\ 3\end{array}\right]= \left[\begin{array}{c}4 \\ -3\end{array}\right]$$
gives 4 equations for a, b, c, d. I get
$$A= \left[\begin{array}{cc}\frac{-7}{25} & \frac{24}{25} \\ \frac{14}{25} & \frac{7}{25}\end{array}\right]$$
for the first reflection.

You can do the same for the second reflection and, of course, their composition is the product of the matrices.

I don't believe
[(333/6350) (644/6350)
(-644/6350) (333/6350)]

is correct because its determinant is not 1, which must be true for a rotation matrix.

 

[lubricarret]

Hi,

Thanks again HallsofIvy.

I used the unit vectors in my formula, and it seems to come out with the same answer; except I tried the technique you gave and I still come up with
[(-7/25) (24/25)
(24/25) (7/25)]
for the second Matrix.

I guess I'm making some calculation error, as
[(333/6350) (644/6350)
(-644/6350) (333/6350)]
Is the matrix I get from the product AB...

Thanks!