06.28.05

Determinants and the cross product

Posted in Mathematics at 10:27 pm by Haris

I haven’t been talking about math yet in this blog, so now it’s time to start. Here’s an interesting little tidbit I realized during teaching this week. I was telling my students about the cross product of vectors, and defined it as a determinant:

a\times b=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_1&a_2&a_3\\ b_1&b_2&b_3 \end{vmatrix}

where as usual \mathbf{i},\mathbf{j},\mathbf{k} are the standard unit vectors in 3-space. Now, the cool thing about this definition is that it shows immediately why a\times b is perpendicular to both a and b. For that, one simply has to check that the dot product of it with each of them is 0. But the dot product with, say, a amounts to computing the determinant:

\begin{vmatrix}a_1&a_2&a_3\\ a_1&a_2&a_3\\ b_1&b_2&b_3 \end{vmatrix}

which of course is zero since two rows of the determinant are the same. Now, one my students asked me: Well, what about other dimensions? I was about to give them the usual answer about there being a mysterious reason why this would work only in dimension 3, but then I realized that in fact this works in all dimensions. If you have n-1 vectors in dimension n, then you can immediately produce a vector perpendicular to them as an n by n determinant:

\begin{vmatrix}\mathbf{e_1}&\mathbf{e_2}&\mathbf{e_3}&\cdots&\mathbf{e_n}\\ a_{1,1}&a_{1,2}&a_{1,3}&\cdots&a_{1,n}\\ a_{2,1}&a_{2,2}&a_{2,3}&\cdots&a_{2,n}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ a_{n-1,1}&a_{n-1,2}&a_{n-1,3}&\cdots&a_{n-1,n} \end{vmatrix}

The exact same reasoning shows that this vector is perpendicular to all the n-1 vectors comprising the rows of this determinant. An amusing special case is when n=2, in which case we get that the vector in the plane perpendicular to the vector \langle a,b\rangle is none other than \begin{vmatrix}i&j\\ a&b\end{vmatrix}=\langle b,-a\rangle! So clearly the magic of the cross product is not simply in it being perpendicular to the original vectors.

It’s stuff like that that makes me love teaching math!

Later