A hyperplane in dimensions is a dimensional subspace. For instance, a hyperplane in $-$dimensional space can be any line in that space and a hyperplane in $-$dimensional space can be any plane in that space. A hyperplane separates a space into two sides.

In general, a hyperplane in $-$dimensional space can be written as For example, a hyperplane in two dimensions, which is a line, can be expressed as $.$

Using this representation of a plane, we can define a plane given an $-$dimensional vector and offset $.$ This vector and offset combination would define the plane One feature of this representation is that the vector is normal to the plane.

Number of Representations

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Given a $-$dimensional vector and a scalar offset which describe a hyperplane $.$ How many alternative descriptions and are there for this plane $?$

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Orthogonality Check

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To check if a vector is orthogonal to a plane characterized by and $,$ we check whether

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