# How to show u x v in R^n space is orthogonal to u and v?

I had a problem where I showed that $u \times v$ in $R^3$ was orthogonal to $u$ and $v$. I was wondering how I could show it for an $R^n$ space? Like, what is the notation/expression to represent a cross product in an $R^n$ space and how would I show that $n$-number of coordinates cancel out?

Thank-you

mfb
Mentor
How do you define ##u \times v## in R^n?
With reasonable generalizations of the cross-product, you can show it in exactly the same way you do in R^3.

mathwonk
Homework Helper
2020 Award
in n space one defines the cross product of n-1 vectors. It is non zero if and only if they are independent and then it is the oriented orthogonal complementary vector to their linear span, with length given by the volume of the block they span. thus if e1,...,en is an oriented orthonormal basis, the cross product of e1,...,en-1, is en.

Matterwave
Gold Member
in n space one defines the cross product of n-1 vectors. It is non zero if and only if they are independent and then it is the oriented orthogonal complementary vector to their linear span, with length given by the volume of the block they span. thus if e1,...,en is an oriented orthonormal basis, the cross product of e1,...,en-1, is en.
Notice however that this generalization has orthogonality explicitly being part of its definition. So this generalization probably doesn't help the OP except to give him the trivial answer (that it is so by definition).

mathwonk