Question

Hi! Professor Auroux, I am reading section 1 about cross product. Can the definiton be generalized to a higher dimension? Thanks!

Answer 1

From what I know, it is only defined in R^3.

Answer 2

The cross product defined as an operation on two linearly independent vectors producing a vector perpendicular to them only exists in three and seven dimensions. (Why seven? I'm not a mathematician, sorry. :))

However, it does generalize in N-dimensional space as an operation on N-1 linearly independent vectors producing a vector perpendicular to the others. Take a short anecdotal example:

\[a=\left<\enspace 1 \enspace 2 \enspace 3 \enspace 4 \enspace \right> \quad b=\left<\enspace 2 \enspace 0 \enspace 4 \enspace 0 \enspace \right> \quad c=\left<\enspace 1 \enspace 0 \enspace 0 \enspace 2 \enspace \right>\]
Then setting up the "cross product" as the determinant of the matrix with the basis vectors in the first row and the other three vectors in the next three rows yields the following vector

\[a=\left<\enspace 16 \enspace 20 \enspace -8 \enspace -8 \> \right>\]
which is indeed orthogonal to the other two, as verified by the dot product.

Answer 3

Where the resulting vector "a" was of course meant to have some other name. ;-) Oops. Too bad we can't edit responses.

Answer 4

That's amazing while two linearly independent vectors could be cross-producted in seven dimensions!! haha:))