Cross products are areas

S. Gilles

2018-05-23

Cross-products are a staple of doing any kind of geometry in three
dimensions. Given vectors v and w, the cross-product v×w is a vector
perpendicular to both v and w, and its magnitude is the area of the
parallelogram with sides v and w.  You can use the cross product
to calculate sin(θ) (the angle between v and w), to find normal
vectors to a parametrized surface, to obtain plane equations, and
all kinds of other useful things.

Calculating a cross product is typically presented as a bit of a
black box, however. The traditional rule goes as follows:

        Stick v, w, and the “vector” ⟨i, j, k⟩ into a matrix, and
        take the determinant: 

                              ⎜ i j k ⎜
        ⟨a, b, c⟩×⟨d, e, f⟩ = ⎜ a b c ⎜
                              ⎜ d e f ⎜

        But don't think about the matrix you're taking the determinant
        of, because it doesn't make sense.

That last part has always irritated me. Determinants don't simply
show up by coincidence, and for such a geometric object as the cross
product, there should be a geometric meaning of formula. So what's
the matrix?

It's easier to start by reducing the dimension count by one. For a
vector v = ⟨a, b⟩, taking

            ⎡ i j ⎤
        M = ⎣ a b ⎦,

|M| yields ⟨b, -a⟩, a vector perpendicular to v. If we blindly draw
the parallelogram corresponding to M⟨1,0⟩ and M⟨0,1⟩, we get

  o---------o--------------------o
  |(0,a+b)  |(j,a+b)      ······ |(i+j,a+b)
  |         |    B   ··········  |
  |    A    |   ··············   |
  o---------o················    |
  |(0,b)   ··(j,b)··········     |
  |       ·················      |
  |  C   ······|M|········   D   |
  |     ·················        |
  |   ··················         |
  |  ··················o---------o
  | ·············      |(i,a)    |(i+j,a)
  |········       E    |     F   |
  o -------------------o---------o
  (0,0)                 (i,0)     (i+j,0)

Where one axis is, abstractly, “whatever contains 0, i, j, and i+j”.

Area calculations are straightforward, even though the “areas” turn
out to be vectors in ℝ².

Entire : (i+j)·(a+b) = ⟨a+b, a+b⟩
A      : a·j         = ⟨  0,   a⟩
B      : a·i/2       = ⟨a/2,   0⟩
C      : b·j/2       = ⟨  0, b/2⟩
D      : b·j/2       = ⟨  0, b/2⟩
E      : a·i/2       = ⟨a/2,   0⟩
F      : a·j         = ⟨  0,   a⟩

The area of the middle is then

  |M| = Entire - (A + B + C + D + E + F)
      = ⟨a+b, a+b⟩ - ⟨a, 2a+b⟩
      = ⟨b, -a⟩,

which is exactly what we want.

The next thing to do is to understand the strange axis, the one
that contains i and j. We need to be able to add i+j, but we don't
need multiplication (if we were in the n=3 case, setting i·j = k
would give a nice quaterion structure, but it's not necessary).
This implies we should take that coordinate as a free ℝ-module V,
with basis {i, j}. Our parallelogram lies in V⊗ℝ. Our naive picture
of the parallelogram is pretty naive, but reality agrees with the
picture barely enough to compute rectangles and triangles, which
is all we need.

(Aside: V is isomorphic to the vector space ℝ², but drawing the
parallelogram in ℝ²×ℝ ≅ ℝ³ wouldn't work: calculating the area
would, at best, give a real number, and |M| should be a vector.
However, it feels extremely wrong to consider V as anything other
than an ℝ-module: 1.1i should be a meaningful quantity.)

To understand the full space V⊗ℝ, consider a small rectangle

                 o----o (w,a+ε)
                 |    |
                 |    |
           (v,a) o----o

The area is ε(w-v), a vector corresponding to a slight push in the
direction from v to w. If we sum a collection of rectangles, we sum
these slight pushes. Consider, for example, the picture for |M|
with both a and b positive. These pushes generally transform vectors
in the direction of j to vectors in the direction of i, so the
pushes are themselves vectors of the form ⟨ε, -ε⟩. The exact
magnitudes of a and b, as well as the slopes of the parallelogram,
turns this into ⟨b, -a⟩.

I claim without proof that this generalizes: for n-1 vectors in ℝⁿ,
placing them and ⟨e₁, e₂, …, eₙ⟩ in a matrix yields a linear map
ℝⁿ → V⊗ℝⁿ⁻¹, and the ‘area’ of the unit rectangle under that map
is a vector perpendicular to all the input vectors.

Conveniently, the concept of a slight push remains the same. Since
only one coordinate of the parallelogram corresponds to vectors,
we can always interpret a small rectangle as a push from one vector
towards another vector. The area calculations get much messier,
though.

This interpretation of the cross product doesn't seem very useful:
it requires thinking about a strange, high-dimensional space, and
I'm not aware of any use for the map M. It's probably cataloged
somewhere in Muir or other standard reference -- I'd be grateful
if someone could point me to it.