Given that,$a =2 \hat{ i }+\hat{ j }-3 \hat{ k } $
$ b =\hat{ i }-2 \hat{ j }+\hat{ k } $
$c =-\hat{ i }+\hat{ j }-4 \hat{ k }$ and $d =\hat{ i }+\hat{ j }+\hat{ k }$
Now, $a \times b =\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 1 & -3 \\ 1 & -2 & 1\end{vmatrix}$
$= i (1-6)-\hat{ j }(2+3)+\hat{ k }(-4-1) $
$a \times b =-5 \hat{ i }-5 \hat{ j }-5 \hat{ k } $
Now, $c \times d =\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ -1 & 1 & -4 \\ 1 & 1 & 1\end{vmatrix}$
$=\hat{ i }(1+4)-\hat{ j }(-1+4)+\hat{ k }(-1-1)$
$=5 \hat{ i }-3 \hat{ j }-2 \hat{ k }$
Now, $( a \times b ) \times( c \times d )=\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ -5 & -5 & -5 \\ 5 & -3 & -2\end{vmatrix}$
$=\hat{ i }(10-15)-\hat{ j }(10+25)+\hat{ k }(+15+25)$
$=-5 \hat{ i }-35 \hat{ j }+40 \hat{ k }=5(-\hat{ i }-7 \hat{ j }+8 \hat{ k })$
$|( a \times b ) \times( c \times d )|=5 \sqrt{(-1)^{2}+(-7)^{2}+(8)^{2}}$
$=5 \sqrt{1+49+64}=5 \sqrt{114}$