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Is cross product defined for vectors of more than three variables

kuuhaku
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  • The answer is NO! – user66407 Oct 03 '15 at 15:35
  • Please google it. – Yes Oct 03 '15 at 15:37
  • Not really, but have a look at this page: https://en.wikipedia.org/wiki/Cross_product and check the subsections "Cross product as an exterior product" and "Multilinear algebra" which indicate how one may look at this in higher dimensions. – Thomas Oct 03 '15 at 15:40
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    @vishnuvardhanreddykavali Not really. You can always take the "product" of $n-1$ vectors in $\Bbb R^n$, but even in $\Bbb R^3$ the result isn't really a vector. The cross product just isn't a very good product. The exterior (wedge) product does about the same thing as the cross product in $\Bbb R^3$ but has much nicer properties (including working on two vectors in $\Bbb R^n$ for any $n$). Look into it. –  Oct 03 '15 at 15:43
  • Yes and no. The cross product really comes from taking the imaginary part of the product of pure imaginary quaternions. This process can be repeated in $\Bbb R^7$ by way of the octonions. – Cameron Williams Oct 03 '15 at 15:45
  • But in $\Bbb R^7$, there's no accepted standard cross product. You have to make an arbitrary choice of one of the $480$ possible cross products. –  Oct 03 '15 at 15:47
  • @Bye_World True that. – Cameron Williams Oct 03 '15 at 15:57

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The cross product can be written formally as $$ \mathbf{x} \times \mathbf{y} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{vmatrix}. $$ There is a unique way to extend this to $n-1$ vectors in $n$ dimensions: $$ \times(\mathbf{x}^{(1)} , \dotsc, \mathbf{x}^{(n-1)}) = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \cdots & \mathbf{e}_n \\ x_1^{(1)} & x_2^{(1)} & \cdots & x_n^{(1)} \\ x_1^{(2)} & x_2^{(2)} & \cdots & x_n^{(2)} \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{(n-1)} & x_2^{(n-1)} & \cdots & x_n^{(n-1)} \end{vmatrix}; $$ this still has the orthogonality and anti-commutativity of the normal cross product.

There is also a (non-unique) way to define a seven-dimensional cross product.

Chappers
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