Chapter 1: Determinants and Exterior Algebra

1. Limits of the cross product

1.1 The cross product is 3D-only

As shown on the page on shear transformations, in 3 dimensions the volume of a parallelepiped can be computed using the cross product (vector product):

However, the cross product $\boldsymbol{b} \times \boldsymbol{c}$ has a fundamental problem:

Why? The cross product takes two vectors $\boldsymbol{b}, \boldsymbol{c}$ and produces a vector orthogonal to both.

• 2D: a direction perpendicular to two vectors leaves the plane (no such direction exists in 2D).
• 4D and above: there are multiple directions perpendicular to two vectors (no unique choice).
• 3D: there is exactly one perpendicular direction (unique up to sign).

So the cross product depends on a "lucky coincidence" of three dimensions.

1.2 The required generalization

We want to compute "the volume of a hyperparallelepiped spanned by $n$ vectors" in $n$ dimensions. We need an operation that:

• works in arbitrary dimension,
• preserves the orientation (sign) of the volume,
• naturally expresses the alternating property of the determinant (sign flip when two vectors are swapped).

This is achieved by the wedge product and exterior algebra.

2. Definition of the wedge product

2.1 Basic properties

The wedge product $\wedge$ takes vectors to higher-order objects. Its most important property is:

An important consequence is obtained immediately:

That is, the wedge product of a vector with itself is 0.

2.2 Geometric meaning

$\boldsymbol{a} \wedge \boldsymbol{b}$ represents the parallelogram (oriented area element) spanned by $\boldsymbol{a}$ and $\boldsymbol{b}$.

Key points:

• $\boldsymbol{a} \wedge \boldsymbol{b}$ is not a vector but a 2-form (a bivector).
• Its "magnitude" equals the area of the parallelogram.
• Antisymmetry encodes orientation as well.

2.3 Higher-order wedge products

The wedge product of three or more vectors is defined analogously:

This is a 3-form representing the parallelepiped (oriented volume element) spanned by $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$.

More generally, in $n$-dimensional space the wedge product of $n$ vectors $\boldsymbol{v}_1, \ldots, \boldsymbol{v}_n$:

is an $n$-form representing the "oriented volume" of the hyperparallelepiped they span.

3. Computation in components

3.1 Standard basis and the wedge product

Consider the standard basis $\boldsymbol{e}_1, \boldsymbol{e}_2$ in 2 dimensions. By antisymmetry:

Therefore the only basis element of 2-forms is $\boldsymbol{e}_1 \wedge \boldsymbol{e}_2$.

3.2 Computation in 2D

For $\boldsymbol{a} = a_1 \boldsymbol{e}_1 + a_2 \boldsymbol{e}_2$ and $\boldsymbol{b} = b_1 \boldsymbol{e}_1 + b_2 \boldsymbol{e}_2$:

The coefficient $a_1 b_2 - a_2 b_1$ is exactly the 2×2 determinant!

3.3 Computation in 3D

Likewise, computing the wedge product of $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ in 3D gives:

3.4 Generalization to $n$ dimensions

In $n$-dimensional space, let $A = [\,\boldsymbol{v}_1 \mid \boldsymbol{v}_2 \mid \cdots \mid \boldsymbol{v}_n\,]$ be the matrix whose columns are the $n$ vectors $\boldsymbol{v}_1, \ldots, \boldsymbol{v}_n$. Then:

This is the essence of the relation between the wedge product and the determinant.

4. Properties of the determinant explained via the wedge product

4.1 Alternating property

The alternating property of the determinant (the sign flips when two columns are swapped) is obtained directly from the antisymmetry of the wedge product:

Hence the determinant of the matrix with two columns swapped has opposite sign (the same holds for rows).

4.2 Two equal columns give 0

The fact that the determinant is 0 when two columns are equal is immediate from

4.3 Multilinearity

The wedge product is linear in each argument:

This corresponds to the multilinearity of the determinant.

4.4 Unified geometric meaning

With the wedge product, the geometric meaning of the determinant is unified across dimensions:

5. Relation to the cross product in 3D

5.1 The Hodge dual

In 3D, the wedge product $\boldsymbol{b} \wedge \boldsymbol{c}$ (a 2-form) and the cross product $\boldsymbol{b} \times \boldsymbol{c}$ (a vector) are closely related.

The Hodge dual ($\star$) converts a 3D 2-form into a vector:

More concretely, written in components:

The Hodge dual acts as:

• $\star(\boldsymbol{e}_2 \wedge \boldsymbol{e}_3) = \boldsymbol{e}_1$
• $\star(\boldsymbol{e}_3 \wedge \boldsymbol{e}_1) = \boldsymbol{e}_2$
• $\star(\boldsymbol{e}_1 \wedge \boldsymbol{e}_2) = \boldsymbol{e}_3$

Therefore

5.2 Why the cross product is 3D-only

The Hodge dual $\star$ converts a $k$-form into an $(n-k)$-form.

For $n = 3$:

• 2-form → $(3-2) = 1$-form (a vector)

This is precisely why $\boldsymbol{b} \wedge \boldsymbol{c}$ (a 2-form) can be expressed as $\boldsymbol{b} \times \boldsymbol{c}$ (a vector).

However, in $n = 4$:

• 2-form → $(4-2) = 2$-form

The Hodge dual of the wedge product of two vectors is again a 2-form, not a vector. This is the mathematical reason why the cross product exists only in 3 dimensions.

6. Summary