If \(\vec a = \frac{1}{{\sqrt {10} }}\left( {3\hat i + \hat k} \right)\,\,and\,\,\vec b = \frac{1}{7}\left( {2\hat i + 3\hat j - 6\hat k} \right)\) , then the value of \(\left( {2\vec a - \vec b} \right)\,\,\cdot\,\,\left[ {\left( {\vec a \times \vec b} \right) \times \left( {\vec a + 2\vec b} \right)} \right]_{}^{}\) is :
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\(\left( {2\vec a - \vec b} \right) \cdot \left[ {\left( {\vec a \times \vec b} \right) \times \left( {\vec a + 2\vec b} \right)} \right]\)
\( = - \left( {2\vec a - \vec b} \right) \cdot \left[ {\left( {\vec a \times 2\vec b} \right) \times \left( {\vec a + \vec b} \right)} \right]\)
\( = - \left( {2\vec a - \vec b} \right) \cdot \left[ {\left( {\left( {\vec a \times 2\vec b} \right) \cdot \vec b} \right)\vec a - \left( {\left( {\vec a + 2\vec b} \right) \cdot \vec a} \right)\vec b} \right]\)
\( = - \left( {2\vec a - \vec b} \right) \cdot \left[ {\left( {\left( {\vec a \cdot \vec b} \right) + 2\vec b \cdot \vec b} \right)\vec a - \left( {\left( {\vec a \cdot \vec a + 2\vec b \cdot \vec a} \right)\vec b} \right)} \right]\)
\( = - \left( {2\vec a - \vec b} \right) \cdot \left[ {0 + 2\vec a - \left( {0 + \vec b} \right)} \right]\)
\( = - \left( {2\vec a - \vec b} \right) \cdot \left( {2\vec a - \vec b} \right)\)
\( = - {\left( {2\vec a - \vec b} \right)^2} = - 4{\vec a^2} + 4\vec a \cdot \vec b - {\vec b^2}\)
= – 4 + 0 – 1 = – 5
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