Hyperplane normal vector

The coefficients for x and y in each linear equation correspond to the component values in the associated hyperplane normal vector, n,. 

A numerical condition for the rate vector r(C) orientation relative to the current hyperplane H(n, b) may be established by computing the angle, 6, between r(C) and the hyperplane normal vector n. Hence,... 

There are very few examples of scalar-mixing cases for which an explicit form for (e, 0) can be found using the known constraints. One of these is multi-stream mixing of inert scalars with equal molecular diffusivity. Indeed, for bounded scalars that can be transformed to a mixture-fraction vector, a shape matrix can be generated by using the surface normal vector n( ) mentioned above for property (ii). For the mixture-fraction vector, the faces of the allowable region are hyperplanes, and the surface normal vectors are particularly simple. For example, a two-dimensional mixture-fraction vector has three surface normal vectors ... 

Here, n is a vector, which is often termed the normal vector, that is perpendicular to all vectors lying in the hyperplane H(n, b), and b is a scalar quantity defined by... 

An alternative method for computing S exists that relies on computing a basis for the null space of the stoichiometric matrix A. Vectors belonging to null (A ) are normal to S, and these vectors may be related to bounding hyperplane constraints that define S. This method is described in Feinberg (1987,2000a, 2000b). 

The first order reliability method (FORM) is an approximate method for assessing the reliability of a structural system. Its basic assumption is to approximate the limit state function (g(Tj,g(x)) = 0) of the structural reliability problem by means of a hyperplane which is orthogonal to the design point vector note that this approximation is constructed in the standard normal space. Thus, the failure probability can be estimated using the Euclidean norm of X, i.e. 

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