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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