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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 ...
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 formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers.
Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem , is named after the Greek mathematician Pappus of Alexandria (4th century AD), who discovered it.
An arbitrary quadrilateral and its diagonals. Bases of similar triangles are parallel to the blue diagonal. Ditto for the red diagonal. The base pairs form a parallelogram with half the area of the quadrilateral, A q, as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A ...
green area = blue area Construction for proof of parallelogram generalization. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the ...
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]
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 can be used to represent two linear maps: one that maps the standard basis vectors to the rows of A, and one that maps them to the columns of A.