Given a point x in n-dimensional space and a hyperplane described by θ and θ0, find the signed distance between the hyperplane and x. This is equal to the perpendicular distance between the hyperplane and x, and is positive when x is on the same side of the plane as θ points and negative when x is on the opposite side.

Question

(Enter theta_0 for the offset θ0.
Enter norm(theta) for the norm ∥θ∥ of a vector θ.
Use * to denote the dot product of two vectors, e.g. enter v*w for the dot product v⋅w of the vectors v and w. )

Answer 1

The signed distance between the hyperplane and point x can be calculated using the formula:

Signed distance = (θ * x + θ0) / ∥θ∥

where θ is the normal vector to the hyperplane, θ0 is the offset of the hyperplane, x is the n-dimensional point, and ∥θ∥ is the norm (length) of the vector θ.

So, to find the signed distance, you would substitute the corresponding values for θ, θ0, and x into the formula:

Signed distance = (θ * x + θ0) / ∥θ∥