Let → u = ^ i - ^ j - 2 ^ k , → v = 2 ^ i + ^ j - ^ k , → v · → w =...

Given:

$\overset{\to}{v} \times \overset{\to}{w} = \overset{\to}{u} + \lambda\overset{\to}{v} . . . . \left(1\right)</p><p>Taking dot product$ with \overset{\to}{w}$, we get</p><p>$\overset{\to}{w} \cdot \left(\overset{\to}{v} \times \overset{\to}{w}\right) = \overset{\to}{w} \cdot \overset{\to}{u} + \lambda\left(\overset{\to}{w} \cdot \overset{\to}{v}\right)

$⇒ \overset{\to}{u} \cdot \overset{\to}{w} + \lambda\left(\overset{\to}{w} \cdot \overset{\to}{v}\right)$ = 0$</p><p>$⇒ \overset{\to}{u} \cdot \overset{\to}{w} = - 2\lambda . . . . \left(2\right)$</p><p>Taking dot product with$\overset{\to}{v} in \left(1\right), we get

$\overset{\to}{v} \cdot \left(\overset{\to}{v} \times \overset{\to}{w}\right)$ = \overset{\to}{v} \cdot \overset{\to}{u} + \lambda\left(\overset{\to}{v} \cdot \overset{\to}{v}\right)$</p><p>$⇒ \left(2 - 1 + 2\right) + \lambda\left(6 + 1 + 1\right) = 0$</p><p>$⇒ \lambda = - \dfrac{1}{2}$</p><p>So, by \left(2\right)$, we get

$\overset{\to}{u} \cdot \overset{\to}{w} = - 2\lambda = 1$