# Generalizing Cross Product

 P: 7 I'm taking multivariate calculus and my teacher just introduced the concept of cross products a week ago. Reading the Wikipedia page, I see that cross products only work in three and seven dimensions, which is puzzling. One use of the cross product for our class is to find the vector orthogonal to the 2 given vectors. My question is can this be generalized to n dimensions to find the vector orthogonal to the n-1 given vectors? Also what is the formal method/operation of doing this? For example, given $u = \left(1,0,0,0\right)$, $v = \left(0,1,0,0\right)$, $w = \left(0,0,1,0\right)$, the vector orthogonal to u, v, and w is given by: $$\left|\begin{array}{cccc} e_{1} & e_{2} & e_{3} & e_{4} \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right| = e_{4}$$ I read a bit about Hodge duality, exterior products, and k-vectors. Much of it was confusing, so could you clarify if you use them as I have little background in linear algebra or tensor theory.
 Mentor P: 11,576 Consider a nxn-matrix where the first n-1 columns (or rows) are filled with n-1 vectors. Now, for each entry in the remaining column (or row), use the determinant of the (n-1)x(n-1)-matrix you get by removing the last column (or row) and the row (or column) your entry is in. This might look complicated, but it is easy to show for the conventional cross-product: $$\begin{pmatrix} a_1 & b_1 &|& \color{red}{a_2b_3-b_2a_3}\\ \color{red}{a_2} & \color{red}{b_2} &|& a_3b_1-b_3a_1 \\ \color{red}{a_3} & \color{red}{b_3} &|& a_1b_2-b_1a_2 \end{pmatrix}$$ It gives a vector which is orthogonal to all other n-1 vectors.