Search results
Results from the WOW.Com Content Network
The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped (see proof). Surface area [ edit ]
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram are incorrect.) In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented ...
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
The proof for Cramer's rule uses the ... the determinant of n vectors of length n will give the volume of the parallelepiped determined by those vectors in the ...
This is a list of volume formulas of basic shapes: [4]: 405–406 ... Parallelepiped – , where , , and are the ...
For a given lattice , this volume is the same (up to sign) for any basis, and hence is referred to as the determinant of the lattice () or lattice constant (). The orthogonality defect is the product of the basis vector lengths divided by the parallelepiped volume;
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
The volume of this parallelepiped is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3. Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by A.