QJEEAdv11Mad3ad: JEE Advanced 2011 Physics - Class 11 JEEAdv11Mad3ad Solution: Let and be three vectors. A vector in the plane of and whose…

We have,

$\overrightarrow v = \lambda \overline a + \mu \overline b$

$= \lambda (\widehat i + \widehat j + \widehat k) + \mu (\widehat i - \widehat j + \widehat k)$

Projection of $\overrightarrow v$ on $\overline c$

${{\overline v \,.\,\overline c } \over {\left| {\overline c } \right|}} = {1 \over {\sqrt 3 }}$

$\Rightarrow {{\left[ {(\lambda + \mu )\widehat i + (\lambda - \mu )\widehat j + (\lambda + \mu )\widehat k} \right]\,.\,\left( {\widehat i - \widehat j - \widehat k} \right)} \over {\sqrt 3 }} = {1 \over {\sqrt 3 }}$

$\Rightarrow \lambda + \mu - \lambda + \mu - \lambda - \mu = 1 \Rightarrow \mu - \lambda = 1 \Rightarrow \lambda = \mu - 1$

$\overline v = (\mu - 1)(\widehat i + \widehat j + \widehat k) + \mu (\widehat i - \widehat j + \widehat k) = \mu (2\widehat i + 2\widehat k) - \widehat i - \widehat j - \widehat k$

$\overline v = (2\mu - 1)\widehat i - \widehat j + (2\mu - 1)\widehat k$

At $\mu = 2$ , $\overline v = 3\widehat i - \widehat j + 3\widehat k$ .