# Exterior algebra

## Exterior algebra

### The exterior (wedge) product

The exterior product of two vectors is:

$$u\land v$$

### The exterior product is anticommutative

This is anticommutative (alternating).

$$u\land v=-v\land u$$

This implies that:

$$u\land u=0$$

### The exterior product is distributive

$$(a+b)\land (c+d)=(a\land c)+(a\land d)+(b\land c)+(b\land d)$$

### Expanding the exterior product of two vectors

We can look at the exterior product in component-basis terms.

Consider $$2$$-dimenional vector space with the following vectors:

$$u=ae_1+be_2$$

$$v=ce_1+de_2$$

The exterior product is:

$$u\land v=(ae_1+be_2)\land (ce_1+de_2)$$

$$u\land v=(ae_1\land ce_1)+(ae_1\land de_2)+(be_2\land ce_1)+(be_2\land de_2)$$

$$u\land v=ac(e_1\land e_1)+ad(e_1\land e_2)+bc(e_2\land e_1)+bd(e_2\land e_2)$$

$$u\land v=ad(e_1\land e_2)-bc(e_1\land e_2)$$

$$u\land v=(ad-bc)(e_1\land e_2)$$

### Exterior (Grassman) algebra

The exterior algebra is the algebra generated by the wedge product.

The term $$u\land v$$ can be interpreted as the area covered by the parallelogram generated by $$u$$ and $$v$$.

As $$a\mathbf u\land b\mathbf v=ab \mathbf u\land \mathbf v$$, we can see that scaling the length of one of the vectors by a scalar, we also increase the exterior product by the same scalar.

### Orientation

We can describe the exterior product of two vectors as $$\mathbf u\land \mathbf v$$ or $$\mathbf v \land \mathbf u$$.