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

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

#### 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