Area & Volume

My previous blog was on two lines intersecting and the intersection gives us a notion of inner product, a peculiar way to multiply two vectors. When we consider area of a two-dimensional plane, we need to define an outer product, which is another way to multiply two vectors.

An area of a section in a flat two-dimensional plane is shown in the diagram below.

When you work out the algebra for sinγ using the identity sin²γ + cos²γ = 1, where cosγ is obtained from the inner product expression, you will find the following result when a and b are two-dimensional vectors.

So, for sinγ we obtain an interesting expression: a₁b₂ − a₂b₁. I said interesting because the indices 1 and 2 corresponding to the first and second components of the vectors a and b are interchanged and for some reason a minus sign is produced. If we stay in 2D, we won't be able to see the beauty of this interesting expression.

To move into a three-dimensional space means we are concerned with volume. If we consider a parallelepiped volume below, we can define three vectors: a = (a₁,a₂,a₃); b = (b₁,b₂,b₃); c = (c₁,c₂,c₃), which form a frame for the parallelepiped as shown below.

Things get truly interesting now. We must extend by analogy the notion of inner product in 2D space to a 3D space:

a⋅b = a_j b_j = a₁b₁ + a₂b₂ + a₃b₃

The index j is repeated, and this implies that j runs from 1 to 3 and the terms are added. Using this definition, it can be shown, after some algebra, that

Area = [(a₂b₃ - a₂b₃)² + (a₃b₁ - a₁b₃)² + (a₁b₂ - a₂b₁)²]^1/2.

The area spanned by the two vectors a and b is a length of a vector - let's call it g - given by

g = (a₂b₃ - a₂b₃, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).

g is the most natural (simple) choice for this vector, whose length corresponds to the area, since g is perpendicular to a and b. There are other 5 possibilities but they are not perpendicular to a and b. Another interesting result is that the vector -g (i.e., opposite to g but equal length) is also a solution, that is its length also yields the area spanned by vectors a and b.

The common choice in applied math is to pick g (not -g) as the vector perpendicular to a and b by imposing an 'index permutation' condition. This condition basically says that a term 'a₂b₃' in g will have a positive sign, while a term 'a₃b₂' will instead have a negative sign, and so on. In the former, the positive sign comes up because the index 2 appears before 3, while in the latter 3 appears before 2. To regulate this sign assignment we define an index-permutation coefficient (its fancy name is Levi-Civita symbol).

An outer product of a and b (in that order) is equal to g,

g = a × b,

and the length of g is equal to the area spanned by vectors a and b. If we flip the order of a and b, we get

-g = b × a,

where the negative sign appears because of the reverse index permutation.

Once we understand that an area in 3D space is equal to the length of vector g, the volume of a parallelepiped spanned by vectors a, b, and c can be obtained simply, as shown by the diagram below.

The volume is equal to the inner product of g and c, and since g = a × b, we get the desired result,

Volume = c⋅(a × b).