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Question
If \[\vec{a} = 2 \hat{ i } + 5 \hat{ j } - 7 \hat{ k } , \vec{b} = - 3 \hat{ i } + 4 \hat{ j } + \hat{ k } \text{ and } \vec{c} = \hat{ i } - 2 \hat{ j } - 3 \hat{ k } ,\] compute \[\left( \vec{a} \times \vec{b} \right) \times \vec{c} \text{ and } \vec{a} \times \left( \vec{b} \times \vec{c} \right)\] and verify that these are not equal.
Solution
\[\text{ Given } : \]
\[ \vec{a} = 2 \hat{ i } + 5 \hat{ j } - 7 \hat{ k }\]
\[ \vec{b} = - 3 \hat{ i } + 4 \hat{ j } + \hat{ k } \]
\[ \vec{c} = \hat{ i } - 2 \hat{ j } - 3 \hat{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 5 & - 7 \\ - 3 & 4 & 1\end{vmatrix}\]
\[ = \left( 5 + 28 \right) \hat{ i } - \left( 2 - 21 \right) \hat{ j } + \left( 8 + 15 \right) \hat{ k } \]
\[ = 33 \hat{ i } + 19 \hat{ j }+ 23 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \vec{b} \right) \times \vec{c} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 33 & 19 & 23 \\ 1 & - 2 & - 3\end{vmatrix}\]
\[ = \left( - 57 + 46 \right) \hat{ i } - \left( - 99 - 23 \right) \hat{ j } + \left( - 66 - 19 \right) \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \vec{b} \right) \times \vec{c} = - 11 \hat{ i } + 122 \hat{ j } - 85 \hat{ k} . . . (1)\]
\[ \therefore \vec{b} \times \vec{c} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ - 3 & 4 & 1 \\ 1 & - 2 & - 3\end{vmatrix}\]
\[ = \left( - 12 + 2 \right) \hat{ i } - \left( 9 - 1 \right) \hat{ j } + \left( 6 - 4 \right) \hat{ k } \]
\[ = - 10 \hat{ i } - 8 \hat{ j }+ 2 \hat{ k } \]
\[ \Rightarrow \vec{a} \times \left( \vec{b} \times \vec{c} \right) = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat { k } \\ 2 & 5 & - 7 \\ - 10 & - 8 & 2\end{vmatrix}\]
\[ = \left( 10 - 56 \right) \hat{ i } - \left( 4 - 70 \right) \hat{ j } + \left( - 16 + 50 \right) \hat{ k } \]
\[ \Rightarrow \vec{a} \times \left( \vec{b} \times \vec{c} \right) = - 46 \hat{ i } + 66 \hat{ j } + 34 \hat{ k } . . . (2)\]
\[\text{ From (1) and (2), we get } \]
\[\left( \vec{a} \times \vec{b} \right) \times \vec{c} \neq \vec{a} \times \left( \vec{b} \times \vec{c} \right)\]
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