Linear algebra - How do I calculate volume from vectors?

Question

How do I calculate the volume of a solid based on vectors in R3?
Given the vectors [1,1,3] [1,2,-1] and [1,4,1] , what is the volume of the polyhedron?

Answer 1

You can calculate the volume of a parallelpiped defined by the three vectors of adjacent edges.
Use the triple product.

A = <1, 1, 3>
B = <1, 2, -1>
C = <1, 4, 1>

These vectors represent three adjacent edges all starting from the same vertex (0, 0, 0).

Let V = volume

V = A • (B X C) = <1, 1, 3> • (<1, 2, -1> X <1, 4, 1>)

V = <1, 1, 3> • <6, -2, 2> = 6 - 2 + 6 = 10

The volume of the parallelpiped is 10.

Answer 2

simulating two-phase flows, interface normal vectors and curvatures are needed for modeling surface tension. In the traditional approach, these quantities are calculated from the spatial derivatives of a scalar function (e.g. the volume-of-fluid or the level set function) at any instant in time. The orders of accuracy of normals and curvatures calculated from these functions are studied. A new method for calculating these quantities is then presented, where the interface unit normals are advected along with whatever function represents the interface, and curvatures are calculated directly from these advected normals. To illustrate this new approach, the volume-of-fluid method is used to represent the interface and the advected normals are used for interface reconstruction. The accuracy and performance of the new method are demonstrated via test cases with prescribed velocity fields. The results are compared with those of traditional approaches.

Answer 3

well just calculate the scalaric size of each of the vectors (length that is in this case) by doing sqrt(a^2+b^2+c^2) then divide your wanted volume into easier shapes like pyramids or boxes and just do it