Question

Consider the hyperplane H=H(1/√3, 1/√3, 1/√3)1​ in R³. Give an explicit description of the mirror MH​ as a mapping in AGO₃ (i.e. find an orthogonal 3-dimensional matrix A and a vector b so that MH​ is given by x→Ax+b

Answer 1

The mirror MH as a mapping in AGO₃ is given by the orthogonal matrix A = (-1/√3, -1/√3, -1/√3) and the vector b = 0. The mirror can be represented as the formula MH(x) = x - 2(x · n)n, where x is a point in AGO₃ and n is the normal vector of the mirror.

Step-by-step explanation:

To find the mirror MH as a mapping in AGO₃, we need to find an orthogonal 3-dimensional matrix A and a vector b. In this case, the hyperplane H is given by H(1/√3, 1/√3, 1/√3). The normal vector n = (1/√3, 1/√3, 1/√3) represents the direction of the mirror. To find the mapping MH, we can use the formula MH(x) = x - 2(x · n)n, where x is a point in AGO₃, · represents dot product, and n is the normal vector.

Let's calculate MH(x) step by step. For any given point x = (x₁, x₂, x₃) in AGO₃:

• Calculate (x · n) = x₁/√3 + x₂/√3 + x₃/√3
• Calculate 2(x · n) = 2(x₁/√3 + x₂/√3 + x₃/√3)
• Calculate 2(x · n)n = (2/√3)(x₁, x₂, x₃)
• Calculate MH(x) = x - 2(x · n)n = (x₁, x₂, x₃) - (2/√3)(x₁, x₂, x₃) = (-x₁/√3, -x₂/√3, -x₃/√3)

Therefore, the mirror MH is given by the orthogonal matrix A = (-1/√3, -1/√3, -1/√3) and the vector b = 0.