Form an $n \times (n-1)$ matrix with columns $v_1, \ldots, v_{n-1}$. Write it as $\pmatrix{w\cr W\cr}$ where $w$ is the first row and $W$ is $(n-1) \times (n-1)$. Correspondingly let $u^T = (a, b)$. Then we need $a w + b W = 0$. Assuming $W$ is invertible, we have $b = - a w W^{-1}$, i.e. $u^T = a (1, -w W^{-1})$. Since we want $u$ to be a unit vector, we take $|a| = \sqrt{1 + \|w W^{-1}\|^2}$. Adding $\Delta$ to $v_{11} = w_1$ subtracts the first row of $W^{-1}$ from $-w W^{-1}$; we then have to adjust $a$ to maintain $u$ as a unit vector.