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If → a = 3 ^ I − ^ J − 2 ^ K and → B = 2 ^ I + 3 ^ J + ^ K , Find ( → a + 2 → B ) × ( 2 → a − → B ) - Mathematics

Sum
\[\text{ If }  \vec{ a } = 3 \hat{ i }- \hat{ j }  - 2 \hat{ k } \text{  and } \vec{b} = 2 \hat{ i }  + 3 \hat{ j } + \hat{ k }  , \text{ find }  \left( \vec{a} + 2 \vec{b} \right) \times \left( 2 \vec{a} - \vec{b} \right) .\]

 

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Solution

\[\text{ Given } : \]

\[ \vec{a} = 3 \hat{ i } - \hat{ j } - 2 \hat{ k }  \]

\[ \vec{b} = 2 \hat{ i } + 3 \hat{ j } + \hat{ k }  \]

\[ \therefore \vec{a} + 2 \vec{b} = 3 \hat{ i } - \hat{ j }  - 2 \hat{ k }  + 2 \left( 2 \hat{ i } + 3 \hat{ j  } + \hat{ k }  \right)\]

\[ = 7 \hat{ i }  + 5 \hat{ j }  + 0 \hat{ k }  \]

\[ \therefore 2 \vec{a} - \vec{b} = 2 \left( 3 \hat{ i }  - \hat{ j }  - 2 \hat{ k  } \right) - \left( 2 \hat{ i }  + 3 \hat{ j  } + \hat{ k  }  \right)\]

\[ = 4 \hat{ i }  - 5 \hat{ j }  - 5 \hat{ k } \]

\[\left( \vec{a} + 2 \vec{b} \right) \times \left( 2 \vec{a} - \vec{b} \right) = \begin{vmatrix}\hat{ i }  & \hat{ j }  & \hat{ k }  \\ 7 & 5 & 0 \\ 4 & - 5 & - 5\end{vmatrix}\]

\[ = \hat{ i } \left( - 25 + 0 \right) - \hat{ j }  \left( - 35 + 0 \right) + \hat{ k }  \left( - 35 - 20 \right)\]

\[ = - 25 \hat{ i }+ 35 \hat { j } - 55 \hat{ k } \]

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APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
Exercise 25.1 | Q 6 | Page 29
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