Question

What is the distance between two parallel hyperplanes a^T x = b1 and x ∈ R^n ? a) |b2 - b1| b) |a^T(b2 - b1)| c) |a^T(b1 - b2)| d) |b1 - b2|

Answer 1

The distance between two parallel hyperplanes defined by equations is |b2 - b1|, assuming the normal vector 'a' is of unit length.

Step-by-step explanation:

The distance between two parallel hyperplanes in R^n given by a^T x = b1 and a^T x = b2 is obtained by taking any point x1 in the first hyperplane and any point x2 in the second hyperplane and projecting the vector x2 - x1 onto the vector a, which is normal to both hyperplanes. Since both points satisfy their respective hyperplane equations, we have a^T x1 = b1 and a^T x2 = b2. The signed distance between x1 and x2 along the direction of a is therefore (a^T x2 - a^T x1) / ||a||. Substituting the values from the hyperplane equations and simplifying gives us the absolute value |b2 - b1| / ||a||. When the normal vector a is of unit length, ||a|| = 1, the distance simplifies to |b2 - b1|, which corresponds to option (d) in the question provided.

Answer 2

The distance between two parallel hyperplanes is computed using the difference in their respective constant terms, divided by the norm of the normal vector to the hyperplanes. The correct formula for the distance is |b2 - b1| / ||a||.

Step-by-step explanation:

The question asks about the distance between two parallel hyperplanes defined as x ∈ R^n and a^T x = b2.

To find the distance, consider a point P1 on the first hyperplane and a point P2 on the second hyperplane such that the line segment P1P2 is perpendicular to both hyperplanes.

The distance d between the hyperplanes is the length of P1P2.

The distance can be calculated using the formula d = |(a^T x2 - b2) - (a^T x1 - b1)| / ||a||, which simplifies to d = |a^T(x2 - x1) - (b2 - b1)| / ||a|| since the hyperplanes are parallel.

This further simplifies to d = |b2 - b1| / ||a||, given that a^T(x2 - x1) = 0 because x2 - x1 is perpendicular to a. Therefore, the correct answer is (a) |b2 - b1| / ||a||.