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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 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 table can be used to calculate the product of any two vectors. ... which is the area of the parallelogram in the plane of x and y with the two vectors as sides. [9]
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 ...
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 following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
Parallelepiped, generated by three vectors A parallelepiped is a prism with a parallelogram as base. Hence the volume V {\displaystyle V} of a parallelepiped is the product of the base area B {\displaystyle B} and the height h {\displaystyle h} (see diagram).
When a vector is divided out of the plane (parallelogram span) formed from it and another vector, what remains is the perpendicular component of the remaining vector, and its length is the planar area divided by the length of the vector that was divided out.