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Question
Find \[\vec{a} . \left( \vec{b} \times \vec{c} \right)\], if \[\vec{a} = 2 \hat {i} + \hat {j} + 3 \hat {k} , \vec{b} = - \hat {i} + 2 \hat {j} + \hat {k}\] and \[\vec{c} = 3 \hat { i} + \hat {j} + 2 \hat {k}\].
Solution
The given vectors are \[\vec{a} = 2 \hat {i} + \hat {j} + 3 \hat {k} , \vec{b} = - \hat {i} + 2 \hat {j} + \hat {k}\] and \[\vec{c} = 3 \hat {i} + \hat {j} + 2 \hat {k}\]
Now,
\[\vec{b} \times \vec{c} = \begin{vmatrix}\hat { i } & \hat {j} & \hat {k}\\ - 1 & 2 & 1 \\ 3 & 1 & 2\end{vmatrix} = 3\hat { i} + 5 \hat {j} - 7 \hat {k}\]
\[\therefore \vec{a} . \left( \vec{b} \times \vec{c} \right) = \left( 2 \hat {i} + \hat {j} + 3 \hat {k} \right) . \left( 3 \hat{i} + 5 \hat {j} - 7 \hat {k} \right) = 2 \times 3 + 1 \times 5 + 3 \times \left( - 7 \right) = 6 + 5 - 21 = - 10\]
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