Let $\mathbf{a}=\mathbf{i}-\mathbf{k},\mathbf{b}=x\mathbf{i}+\mathbf{j}+(1-x)\mathbf{k}$ and $\mathbf{c}=y\mathbf{i}+x\mathbf{j}+(1+x-y)\mathbf{k}$ Then [$a b c$ ] depends on

$[\mathbf{a}\mathbf{ }\mathbf{b}\mathbf{ }\mathbf{c}]=\left|\begin{array}{ccc}1& 0& -1\\ x& 1& 1-x\\ y& x& 1+x-y\end{array}\right|$

$=\left|\begin{array}{ccc}1& 0& 0\\ x& 1& 1\\ y& x& 1+x\end{array}\right| \left(\text{ using }{C}_3\to C_3+C_1\right)$

   $=(1+x)-x=1$ which depends neither on 

$x\text{ nor on }y\text{. }$