The result of reflecting across the line y = −x and then

Question

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?

Answer 1

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