The cross products from two to four dimensions, expressed in determinant form and long form.

Here is the two dimensional "cross product". It is an operation on a single vector, and it simply rotates the vector by 90 degrees (so that it is perpendicular). Perhaps flatlander physicists have a use for this sort of thing.

```cross u = | i  j  | = u2 i - u1 j
| u1 u2 |

```
Next, the three dimensional cross product that you may be familiar with. Two vectors in, one vector out. It is perpendicular to both input vectors.
```                       | i  j  k  |
cross (u, v) = u x v = | u1 u2 u3 |
| v1 v2 v3 |

= (u2 v3 - u3 v2) i - (u1 v3 - u3 v1) j + (u1 v2 - u2 v1) k

```
Then ... we have the four dimensional cross product. Yes, that gigantic block of symbols in your lower peripheral vision. Three vectors in. Single output vector, which is this time perpendicular to all three input vectors. More interesting things: swap two of the parameters, the result is negated. Negate a parameter, the result is negated.

Certain optimisations can be made when computing this. The lower 2x2 determinants (eg. v3 w4 - v4 w3) are each used twice, so they can be computed once and then remembered the second place. Still, it is a hefty job.

(I hope this works out right! *crosses fingers*)

```                  | i  j  k  l  |
cross (u, v, w) = | u1 u2 u3 u4 |
| v1 v2 v3 v4 |
| w1 w2 w3 w4 |

/                                                            \
=   | (v3 w4 - v4 w3) u2 - (v2 w4 - v4 w2) u3 + (v2 w3 - v3 w2) u4 | i
\                                                            /

/                                                            \
- | (v3 w4 - v4 w3) u1 - (v1 w4 - v4 w1) u3 + (v1 w3 - v3 w1) u4 | j
\                                                            /

/                                                            \
+ | (v2 w4 - v4 w2) u1 - (v1 w4 - v4 w1) u2 + (v1 w2 - v2 w1) u4 | k
\                                                            /

/                                                            \
- | (v2 w3 - v3 w2) u1 - (v1 w3 - v3 w1) u2 + (v1 w2 - v2 w1) u3 | l
\                                                            /

```
Oh, you want to see the five dimensional cross product in long form? Write it out yourself, you sick freak! :-) (it should be five times as big as the four dimensional one)

And that is why determinants are convenient.