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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).
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 right-hand side is the Gram determinant of a and b, the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle θ between the two vectors, as:
A parallelepiped is a region bounded by three pairs of ... or area. The Euclidean plane ... The vectors v and w can be visualized as vectors starting at r 0 and ...
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.
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 magnitude of u ∧ v is the area of this parallelogram. Notice that because V has dimension two the basis bivector e 1 ∧ e 2 is the only multivector in ΛV. The relationship between the magnitude of a multivector and the area or volume spanned by the vectors is an important feature in all dimensions.
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 ...