Solutions To D.E. Littlewood Non-Commutative Algebra Problems

 1) Consider the parallelopipped  below spanned by the vectors u, v and w in 3-space.

Find the volume of this solid given that:

u = (1, 1, 3)

v = (1, 2, -1)

w = (1, 4, 1)

Solution:

We let the volume be Vol(u, v, w) and:

Vol (u, v, w) = Det [u, v, w]

Then we need:

=

    =      [2 -(-4) - (1 - (-1)) + 3(4 - 2)] = 6 - (2) + 3(2)

Vol (u, v. w) = 12 - 2 = 10 cubic units

2)Show how the Grassman algebra formula for a single component of volume element  k e₁ e ₂  e ₃  can be used to find the volume of a tetrahedron.

Find that volume for this tetrahedron, given the concurrent vectors are:

a = (3, 0, 0)

b = (1, 4, 0)

c = (2, 1, 5)

Solution:

In Grassman algebra, the fundamental volume element V is represented by the wedge product of three basis vectors:

Then the product of the three concurrent vectors given (a,b,c) is the exterior product:

Then using the anticommutative property (identified in blog post):

The product of the linear combinations can then be evaluated:

Where det (a,b,c,) is the determinant of the vector components:

This scalar then gives the volume spanned by the vectors, much like the scalar in #1 (det) gave the volume for the parallelopiped.

Let D =   det (a,b,c), then:

Volume of tetrahedron = 

 1/6 det (a,b,c) = D/ 6  = 60/ 6 = 10 cubic units