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