Let $A$ be an $n-1 \times n$ matrix. The span of the rows of $A$ define a hyperplane in $\mathbb{R}^n$; let $u$ be the unit normal to this hyperplane.
Now, let $x \in \mathbb{R}^{n-1}$, and replace each element $A_{i1}$ in the first column of $A$ with a variable term $A_{i1}cos(\theta) - x_i sin(\theta)$; let $u(\theta)$ be the resulting unit normal (thus, $u(0) = u$).
Suppose we slide $\theta$ up from $0$ continuously, and stop if/when $u_k(0) - u_k(\theta) = \Delta$ for some index $k \in \{1, \dots, n\}$. In terms of $A$, $x$, $k$, and $\Delta$, how far can we slide $\theta$?
EDIT: I've managed to reduce the problem to a possibly simpler one. If we delete column $k$ from the modified matrix, then can we find an analytic form for its determinant? In other words, what is $\det(A_{*,\sim k}, \theta)$?