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.
(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. )
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∥
To find the signed distance between the hyperplane and the point x, you can use the formula:
distance = (θ * x + θ0) / ∥θ∥
Let me break down the steps for you:
1. Compute the dot product of θ and x, denoted as θ * x.
2. Add θ0 to the result from step 1.
3. Divide the result from step 2 by the norm of θ, denoted as ∥θ∥.
The final value will give you the signed distance between the hyperplane and the point x.