# If $\overrightarrow a=\hat i-\hat k,\:\overrightarrow b=x\hat i+\hat j+(1-x)\hat k\:and\:\overrightarrow c=y\hat i+x\hat j+(1+x-y)\hat k$, then $[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$ depends on?

$\begin{array}{1 1} x\;alone \\ y\;alone \\ Both \;x\;and \;y \\ Independent \;of\; x \;and\; y \end{array}$

$[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=\left |\begin{array}{ccc}1 & 0 & -1 \\x & 1 & 1-x\\y & x& (1+x-y)\end {array}\right|$
$=1+x-y-x+x^2-x^2+y=1$