In general, a hyperplane in ndimensional space can be written as 60 l 61ml l 02:02 l - - - + 6,133,, = 0. For example, a hyperplane in two dimensions, which is a line, can be expressed as Aml + 3332 l C = 0. 91 92 Using this representation of a plane, we can define a plane given an ndimensional vector 6 = and offset 60. 9n This vector and offset combination would define the plane 00 + 61331 + 02.732 + - - - + 9,133,, = 0. One feature of this representation is that the vector 0 is normal to the plane. Number of Representations 1 point possible (graded) Given a ddimensional vector 0 and a scalar offset 90 which describe a hyperplane P : 9 - a: + 60 = 0. How many alternative descriptions 0' and 63 are there for this plane 'P? Orthogonality Check 1 point possible (graded) To check if a vector :1: is orthogonal to a plane 'P characterized by 0 and 60, we check whether 0 x=a6forsomeaER Perpendicular Distance to Plane '1 point possible (graded) Given a point x in nudimensional space and a hyperplane described by 9 and 90, find the signed distance between the hyperplane and a}. This is equal to the perpendicular distance between the hyperplane and cc, and is positive when :1: is on the same side of the plane as 9 points and negative when m is on the opposite side. (Enter theta_0 for the offset 00. Enter norm(theta) for the norm \"6\" of a vector 9. Use * to denote the dot product of two vectors, e.g. enter v*w for the dot product '0 ' w of the vectors '0 and w. ) Concept Check 4 points possible (graded) 1. Is the value of fx (:3) always 6 [0, 1]? 0 yes 0 no 6 2. Fora.