$\vec{u}$,$\vec{v}$,$\vec{w} \in \mathbb{C}^{2017}$ are linearly independent

Question

$\vec{u}$,$\vec{v}$,$\vec{w} \in \mathbb{C}^{2017}$ are linearly independent. Find $a \in \mathbb{C}$ such $a\vec{u}+i\vec{v}$, $a\vec{v}+i\vec{w}$, $a\vec{w}+i\vec{u}$ are linearly independent. If we can assume without loss of generality that $\vec{u} = [1, 0, \dots, 0]$, $\vec{v} = [0, 1, \dots, 0]$, $\vec{w} = [0, 0, 1, \dots, 0]$ then $a\vec{u}+i\vec{v} = [a, i, 0 \dots, 0]$, $a\vec{v}+i\vec{w} = [0, a, i, 0 \dots, 0]$, $a\vec{w}+i\vec{u} = [i, 0, a, 0 \dots, 0]$. $det(\begin{pmatrix} a & 0 & i \\ i & a & 0 \\ 0 & i & a \\ \vdots & \vdots & \vdots \\ 0 & 0 & 0 \end{pmatrix}) = a^3-i$, so $a^3 \neq i$. Am I right?