Vector space for cross products?

In summary, the cross product is typically defined for vectors in three dimensions, but there is a more general definition for other dimensions. This definition involves using the "alternating tensor" in m dimensions and taking the determinant of a matrix formed by the n-1 vectors. This gives a linear mapping of a vector, resulting in a unique vector known as the cross product of the first n-1 vectors.
  • #1
pivoxa15
2,255
1
What vector space are cross products done in?
 
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  • #2
If you're talking about the standard definition, it's only defined for vectors in three dimensions, so anything that has three dimensions you could do the cross product in. Why?

It occurs to me that I think there's a more general form for other dimensions, but I don't know it
 
  • #3
A more general definition is this: Let [itex]\epsilon_{ijk...m}[/itex] be the "alternating tensor" in m dimensions: +1 if ijk...m is an even permutation of 123...n, -1 if an odd permutation, 0 otherwise. Then we can define the "cross product" of n-1 vectors [itex]v_1[/itex], [itex]v_2[/itex], ..., [itex]v_{n-1}[/itex] to be the vector [itex]v= \Sigma \epsilon_{ij...m}v_{1i}v_{2j}...v_{n-1,m}[/itex] where the sum is take over repeated indices. If n= 3 then that gives the cross product on R2.
 
  • #4
given n-1 vectors in n dimensions, let them act on another vector w by taking the detrminant of the mtrix the n vectors form togetehjr. that gives alinear \map of w, which is thus dotting with aunique vector caled the cross product of the first n-1 vectors.
 

Related to Vector space for cross products?

1. What is a vector space for cross products?

A vector space for cross products is a mathematical concept that refers to a set of vectors that can be multiplied together using the cross product operation. This operation produces a new vector that is perpendicular to both of the original vectors.

2. How is the cross product operation defined in a vector space?

In a vector space, the cross product operation is defined as the multiplication of two vectors to produce a new vector that is perpendicular to both of the original vectors. This operation follows the right-hand rule and is denoted by the symbol "×".

3. What are the properties of a vector space for cross products?

A vector space for cross products has several properties, including closure, commutativity, and distributivity. Closure means that the result of a cross product is always another vector in the same vector space. Commutativity states that the order of the vectors in the cross product does not matter. Distributivity means that the cross product operation can be distributed over vector addition.

4. How is the cross product used in real-life applications?

The cross product has many applications in physics, engineering, and computer graphics. It is used to calculate torque in physics, determine the direction of force in engineering, and compute the normal vector for lighting and shading in computer graphics.

5. Can a cross product be performed in any vector space?

No, a cross product can only be performed in three-dimensional vector spaces. This is because the cross product requires the existence of a third orthogonal dimension in order to produce a perpendicular vector. It is not defined in two-dimensional or higher-dimensional vector spaces.

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