The result of reflecting across the line y = −x and then
rotating 330 degrees counterclockwise around the origin is an
isometry T . Represent T by a 2 × 2 matrix. There is more than
one way to do it. Use the point (1, 1) to check your answer.
How do I find the rotations?
The first operation in this case is a reflection along the line y = -x. The 2x2 matrix representing this is,
[[ 0, -1],
[-1, 0]]
The next operation is a rotation about the origin by 330 degrees counterclockwise.
The matrix that represents this rotation in 2D space is,
[[cos(330deg), -sin(330deg)],
[sin(330deg), cos(330deg)]]
Plug in the equivalent radians for 330 degrees (330 * pi/180) to find the cos and sin.
Let's refer to the matrices as M1 for reflection and M2 for rotation.
The resulting transformation would be the product of M1 and M2.
To compute this, multiply the two transformation matrices M1 and M2 together:
M1*M2 = [[ cos(330), sin(330)],
[-sin(330), cos(330)]]
The result of this operation, M = M1*M2, is the 2x2 matrix that represents the isometry T.
To check if T is correct, you can try applying it to the point (1,1). You should obtain the same coordinates as when you apply the reflection and the rotation operations to the point, in the same order.
Remember that matrix multiplication corresponds to a composition of transformations, so the order you multiply your matrices in matters – in this case, M = M1*M2 represents the reflection followed by the rotation.
Remember also that to apply a 2x2 matrix to a point (x,y), you treat the point as a column vector and multiply it by the matrix on the left:
[[ a, b], . [x] = [ax + by]
[ c, d]] . [y] = [cx + dy]
So, applying T to a point (x,y), you would perform these calculations to obtain the new coordinates (x',y'):
x' = a*x + b*y
y' = c*x + d*y
To find the rotations in this problem, we need to break down the steps we're taking:
Step 1: Reflecting across the line y = -x.
Step 2: Rotating 330 degrees counterclockwise around the origin.
In Step 1, reflecting across the line y = -x, we can use the following reflection matrix:
R = [[0, -1], [-1, 0]]
This matrix flips the x and y coordinates, effectively reflecting the point across the line y = -x.
In Step 2, rotating 330 degrees counterclockwise around the origin, we can use the following rotation matrix:
θ = 330 degrees in radians = (11π/6) radians
Rot = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]
= [[cos(11π/6), -sin(11π/6)], [sin(11π/6), cos(11π/6)]]
Step 1 and Step 2 can be combined by multiplying the reflection matrix R with the rotation matrix Rot:
T = Rot * R
To find the resulting isometry T, we multiply the rotation matrix with the reflection matrix. You can calculate this by multiplying the corresponding elements of the matrices and summing them up:
T = [[cos(11π/6)*(0) + (-sin(11π/6))*(-1), cos(11π/6)*(-1) + (-sin(11π/6))*(0)], [sin(11π/6)*(0) + cos(11π/6)*(-1), sin(11π/6)*(-1) + cos(11π/6)*(0)]]
= [[sin(11π/6), cos(11π/6)], [-cos(11π/6), sin(11π/6)]]
So, the matrix T representing the isometry obtained by reflecting across the line y = -x and then rotating 330 degrees counterclockwise around the origin is:
T = [[sin(11π/6), cos(11π/6)], [-cos(11π/6), sin(11π/6)]]
To check your answer using the point (1, 1), you can multiply the matrix T by the coordinates of the point and see if it reflects and rotates correctly.