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.
I know how to make the matrices but I don't know how to combine them into one. Do I just add them?
Would you please help me?? Anyone??
no
To represent the combination of reflecting across the line y = x and rotating 330 degrees counterclockwise around the origin as a single 2x2 matrix, you need to multiply the matrices representing the individual transformations. Matrix multiplication is not the same as addition.
Let's first write the matrix for the reflection across the line y = x. Reflection across the line y = x can be represented by the matrix [0 1; 1 0].
Next, we need to write the matrix for the rotation of 330 degrees counterclockwise around the origin. The rotation matrix can be written as [cos(330°) -sin(330°); sin(330°) cos(330°)]. Make sure to use radians instead of degrees when calculating the values for sine and cosine functions.
Now, multiply the two matrices by multiplying corresponding elements and adding the products:
[0 1; 1 0] * [cos(330°) -sin(330°); sin(330°) cos(330°)]
This will give you the combined matrix for the composition of the two transformations.
To combine the reflections and rotations into one matrix, you need to perform matrix multiplication.
Let's start with the individual transformations:
1. Reflection across the line y = x:
When reflecting a point (x, y) across the line y = x, you can swap the x and y coordinates. The matrix representation of this reflection is:
[0 1]
[1 0]
2. Rotation of 330 degrees counterclockwise around the origin:
To rotate a point (x, y) counterclockwise around the origin by an angle θ, you can use the following matrix:
[cos(θ) -sin(θ)]
[sin(θ) cos(θ)]
In this case, the rotation angle is 330 degrees, which is equal to 330 * (π/180) radians. So, the matrix representation of this rotation is:
[cos(330π/180) -sin(330π/180)]
[sin(330π/180) cos(330π/180)]
Now, to combine the two transformations into one matrix, you need to multiply the matrices together in the correct order. Matrix multiplication is not commutative, so the order matters.
In this case, you first apply the reflection and then the rotation, so the combined transformation is the matrix product of the reflection matrix and the rotation matrix:
[0 1] [cos(330π/180) -sin(330π/180)] [? ?]
[1 0] * [sin(330π/180) cos(330π/180)] = [? ?]
You can perform the matrix multiplication to compute the individual elements of the resulting matrix. Once you have the combined matrix, you can use it to transform the point (1, 1) by multiplying it with the matrix:
[? ?] [1]
[? ?] * [1]
By performing this matrix multiplication, you will obtain the transformed coordinates of the point (1, 1).