TITLE: What is the "determinant" of two vectors?
QUESTION [6 upvotes]: I came across the notation $\det(v,w)$ where $v$ and $w$ are vectors. Specifically, it was about the curvature of a plane curve:
$$\kappa (t) = \frac{\det(\gamma'(t), \gamma''(t)) }{\|\gamma'(t)\|^3}$$
What is it supposed to mean?

REPLY [3 votes]: Vectors in a plane $v,w$ can be written as column matrices: $$v=\begin{bmatrix}v_1\\v_2\end{bmatrix}, \ \ \ w=\begin{bmatrix}w_1\\w_2\end{bmatrix}.$$ Put several of such column matrices side by side, and you get a square matrix: $$(v,w)=\begin{bmatrix}v_1&w_1\\v_2&w_2\end{bmatrix}.$$ The determinant $\det(v,w)$ is simply the determinant of this square matrix: $$\det(v,w)=\det\begin{bmatrix}v_1&w_1\\v_2&w_2\end{bmatrix}.$$