vector cross product

[Darkwraith]

Could you do a cross product on two vectors in which each vector contains more than 3 elements?

[TheManWhoCan]

Yes, you can. Supposing you have two 3d vectors,

(a₁,a₂,a₃)
(b₁,b₂,b₃)

The cross product is given by the determinant,

| i     j     k |
| a1    a2    a3 |
| b1    b2    b3 |

In three spatial dimensions, any point can be described by a linear combination of the vectors i, j, and k, which are defined,

i x j = k
j x k = i
k x i = j

If in four dimensions you had an extra unit vector l and defined the four of them as,

i x j = k
j x k = l
k x l = i
l x i = j

Then you could probably represent a cross product of the vectors,

(a₁,a₂,a₃,a₄)
(b₁,b₂,b₃,b₄)
(c₁,c₂,c₃,c₄)

By the determinant,

| i     j     k     l |
| a1    a2    a3    a4 |
| b1    b2    b3    b4 |
| c1    c2    c3    c4 |

By definition the 3d cross product returns a vector orthogonal to both the vectors it's applied to. Since the four unit dimension vectors we defined are mutually orthogonal, it is necessary to apply the 4d cross product to three vectors in order to find a vector orthogonal to all three.

There you go I've just made all of that up. That's how I would imagine a 4d cross product working.

[Acidic]

I've never done one in four, but I'll help you out with one in 3D.

Let's say you have this:

| i  j  k  |
| a  b  c  |
| d  e  f  |

then the product would be:
i(bf-ec)-j(af-dc)+k(ae-db)

[Darkwraith]

So is there a defined way of handling the cross product of 4D vectors?

[Darkwraith]

Ok, I think I got it. The cross product of vectors is basically vector multiplication. So, I just have to multiply them as such:

http://www.ma.umist.ac.uk/servicemat...torproduct.pdf

Or is cross product something special?

[nicktheman]

Um, probably silly question, but how do you get 4d vectors?

(I just can't visualise it)

[Acidic]

no-one can visualise 4 dimensions, but you can still calculate it in. you can calulate in 264 dimensions if you want.

[Darkwraith]

I am trying to create a general Vector class (no, not the container type.) In order to do so, I need to know how to do a cross product for an n-dimension vector.

I found something on vector multiplication, which is different from matrix multiplication. There is no real visualization involved. I am just trying to get the basic mathematical rules about vector multiplication.

[Acidic]

well. I've never done it wit ha 4D vector. So I cant help, sorry.

[azteched]

For a 4-d (on N-d) cross product just extend the determinant as required.

[Darkwraith]

So for vectors with dimensions larger than 3, we will need to do the cross product over several vectors?