VC2.5

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a\cdot b\times c=\begin{vmatrix} t & -3 & 2t \\ 1 & -2 & 2 \\ 3 & t & -1 \end{vmatrix}\,

=\begin{vmatrix} t & 2t-3 & 0 \\ 1 & 0 & 0 \\ 3 & 6+t & -7 \end{vmatrix}\, [Performing C_2 \longrightarrow C_2+2C_1,C_3=C-3-2C_1\,]

=-7[0-(2t-3)]\, [Expanding by third column]

=7(2t-3)\,

Hence,\int_0^2 a\cdot b\times c \,dt=7\int_1^2 (2t-3)\,dt=7[t^2-3t]\, [At t=(1,2)]

=7[(4-6)-(1-3)]=0\,

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