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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 Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square. If m = n, then f is a function from R n to itself and the Jacobian matrix is a ...
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 ...
English: The area of a parallellogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. Date 25 January 2025
The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one. The determinant of a square matrix A (denoted det(A) or | A |) is a number encoding certain properties of the matrix.
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra features. [8] In September 2023, Desmos released a beta for a 3D calculator, which added features on top of the 2D calculator, including cross products, partial derivatives and double-variable parametric equations.