Find a single matrix that defines a rotation of the plane through an angle of π/6 about the origin, while at the same time moves points to five times their original distance from the origin.
To find a single matrix that defines both rotation and scaling, we can combine the transformation matrices for rotation and scaling.
Let's start by finding the matrix for rotation by an angle of π/6 about the origin. This can be represented by the matrix:
R = [[cos(θ), -sin(θ)],
[sin(θ), cos(θ)]]
where θ is the angle of rotation. In this case, θ = π/6, so:
R = [[cos(π/6), -sin(π/6)],
[sin(π/6), cos(π/6)]]
Next, we need to find the matrix for scaling by a factor of 5. This can be represented by the matrix:
S = [[sx, 0],
[0, sy]]
where sx is the scaling factor in the x-direction, and sy is the scaling factor in the y-direction. In this case, we want to scale points to five times their original distance from the origin, so sx = sy = 5:
S = [[5, 0],
[0, 5]]
To combine the rotation and scaling matrices into a single matrix, we can simply multiply them together:
M = RS
M = [[cos(π/6)*5, -sin(π/6)*5],
[sin(π/6)*5, cos(π/6)*5]]
Simplifying the expressions, we get:
M = [[(√3)/2 * 5, -1/2 * 5],
[1/2 * 5, (√3)/2 * 5]]
M = [[(5√3)/2, -5/2],
[5/2, (5√3)/2]]
So, the matrix that defines a rotation of the plane through an angle of π/6 about the origin, while at the same time scaling points to five times their original distance from the origin, is:
M = [[(5√3)/2, -5/2],
[5/2, (5√3)/2]]