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The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. If the matrix entries are real numbers, the matrix A represents the linear map that maps the basis vectors to the columns of A.
The vector area of a parallelogram is given by the cross product of the two vectors that span it; it is twice the (vector) area of the triangle formed by the same vectors. In general, the vector area of any surface whose boundary consists of a sequence of straight line segments (analogous to a polygon in two dimensions) can be calculated using ...
The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule [1] and a magnitude equal to the area of the parallelogram that the vectors span. [2] The cross product is defined by the formula [8] [9]
Unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product. Once we have established a multiplication table, it is then applied to general vectors x and y by expressing x and y in terms of the basis and expanding x × y through ...
Vectors involved in the parallelogram law. In a normed space, the statement of the parallelogram law is an equation relating norms: ‖ ‖ + ‖ ‖ = ‖ + ‖ + ‖ ‖,.. The parallelogram law is equivalent to the seemingly weaker statement: ‖ ‖ + ‖ ‖ ‖ + ‖ + ‖ ‖, because the reverse inequality can be obtained from it by substituting (+) for , and () for , and then simplifying.
The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices. The two-dimensional Euclidean vector space is a real vector space equipped with a basis consisting of a pair of orthogonal unit vectors = [], = [].
The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram. The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the ...