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# Precalculus MM: A New Day, The Ancient Queen: Systems of Linear Equations and Matrices 3: NWN 0151b

The method by which the 30-60-90 feeds images into Zanzi's dreams relies on transmitting data into the area bounded by quadrilateral and triangular polygons. The data are propagated among many billions of these polygons located in neocortex procedural and semantic memory centres during hypnagogic and REM sleep. In order to tailor the data for each polygon the 30-60-90 uses 2x2 determinants to compute the area of the polygon, such that for a quadrilateral area = the absolute value of the determinant whose elements are obtained from the components of two vectors originating from the same vertex. A triangle area = the half the computed determinant quadrilateral area.
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The method by which the 30-60-90 propagates the data to control Zanzi's dreams relies on targeting streams of data from its location in Zanzi's right amygdala to specific memory locations within her neocortex. This requires computation of 3D vectors using a 3x3 determinant matrix array. Unit vectors i hat, j hat and k hat are used in cross product vector computations, as the cross product of 3D vectors is the only product that results is a vector. The cross product is found by setting up a 3x3 determinant and entering the first row elements as the unit vectors i hat, j hat and k hat. For two 3D vectors u and v, u x v means that the second row elements will be the components for vector u and the third row elements will be the components for vector v. (Note that v x u results in a vector of the same size and opposite direction to u x v, so u x v = -(v x u)). Because the dot product of a vector with a cross product that has the same vector as either the multiplicand or multiplier position results in zero, indicating that cross product is perpendicular to the original vector, the 30-60-90 can also use the cross product of vectors to find an equation for a plane and then set up a data stream perpendicular to this plain. Firstly it selects three points A, B, C, say, that lie within and determine the plane of desired orientation for the data stream. Then, from the coordinates of each point it computes the components of two vectors that lie in the plane, d, say vector AB and vector AC. It then places these vectors into unit vector form and then obtains a vector normal to plane d using the cross products of AB x AC using the 3x3 determinant method. The components calculated for the resulting normal vector are then set as the coefficients (a, b and c) of the x, y and z variables of the plane equation: ax + by + cz = d. Finally to find d it chooses the coordinates of one of the points A, B or C that lie the plane.
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So that Zanzi will be able to remember important aspects of the dream when she wakes, the 30-60-90 duplicates essential data for propagation within specific polygons comprising certain hippocampi constructs. As with the transmission of data into the neocortex memory centres, these data are tailored to fill the area bounded by certain quadrilateral and triangular polygons. However in this case the 30-60-90 uses the fact that the area of a quadrilateral can be obtained from the magnitude of the cross products of two vectors because this is equal to the product of the magnitude of each vector and the sine of the angle between the two vectors (theta) when set as a vertex of the quadrilateral, which in turn is derived from the base x height area formula for quadrilaterals, i.e. b.h = |u||v|sin theta = |u x v|. The cross product of vectors u and v is computed using 3D unit vectors as first row elements within a 3x3 matrix. Taking the magnitude of the computed components for i hat, j hat and k hat yields the quadrilateral area. Dividing this area by two yields the triangular area.