Let → a = 3 ^ i + ^ j and → b = ^ i + 2 ^ j + ^ k . Let → c be a...

Given,

\overset{\to}{a} = 3\overset{^}{i} + \overset{^}{j} and \overset{\to}{b} = \overset{^}{i} + 2\overset{^}{j} + \overset{^}{k}

Let $\overset{\to}{c}$ be a vector satisfying $\overset{\to}{a} \times \left(\overset{\to}{b} \times \overset{\to}{c}\right) = \overset{\to}{b} + \lambda\overset{\to}{c}$. If $\overset{\to}{b}$ and $\overset{\to}{c}$ are non-parallel, then the value of $\lambda$ is

Now putting the value of \overset{\to}{a} = 3\overset{^}{i} + \overset{^}{j} \text{ and } \overset{\to}{b} = \overset{^}{i} + 2\overset{^}{j} + \overset{^}{k} in $\overset{\to}{a} \times \left(\overset{\to}{b} \times \overset{\to}{c}\right) = \overset{\to}{b} + \lambda\overset{\to}{c}$ we get,

$⇒ \overset{\to}{a} \times \left(\overset{\to}{b} \times \overset{\to}{c}\right) = \overset{\to}{b} + \lambda\overset{\to}{c}$

$⇒ \overset{\to}{a} \cdot \overset{\to}{c}\left(\overset{\to}{b}\right) - \left(\overset{\to}{a} \cdot \overset{\to}{b}\right)\overset{\to}{c} = \overset{\to}{b} + \lambda\overset{\to}{c}$

On comparing both side we get,

$⇒ \overset{\to}{a} \cdot \overset{\to}{c} = 1 \text{ and } \overset{\to}{a} \cdot \overset{\to}{b} = - \lambda$

⇒ \left(3\overset{^}{i} + \overset{^}{j}\right) \cdot \left(\overset{^}{i} + 2\overset{^}{j} + \overset{^}{k}\right) = - \lambda

$⇒ \lambda = - 5$