Let $\overrightarrow a = \widehat i + \widehat j + 2\widehat k$, $\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k$ and $\overrightarrow c = \widehat i - \widehat j + \widehat k$ be three given vectors. Let $\overrightarrow v$ be a vector in the plane of $\overrightarrow a$ and $\overrightarrow b$ whose projection on $\overrightarrow c$ is ${2 \over {\sqrt 3 }}$. If $\overrightarrow v \,.\,\widehat j = 7$, then $\overrightarrow v \,.\,\left( {\widehat i + \widehat k} \right)$ is equal to :

<p>Let $$\overrightarrow v = {\lambda _1}\overrightarrow a + {\lambda _2}\overrightarrow b $$, where ${\lambda _1},\,{\lambda _2} \in R$.</p> <p>$$ = ({\lambda _1} + 2{\lambda _2})\widehat i + ({\lambda _1} - 3{\lambda _2})\widehat j + (2{\lambda _1} + {\lambda _2})\widehat k$$</p> <p>$\because$ Projection of $\overrightarrow v$ on $\overrightarrow c$ is ${2 \over {\sqrt 3 }}$</p> <p>$\therefore$ $${{{\lambda _1} + 2{\lambda _2} - {\lambda _1} + 3{\lambda _2} + 2{\lambda _1} + {\lambda _2}} \over {\sqrt 3 }} = {2 \over {\sqrt 3 }}$$</p> <p>$\therefore$ ${\lambda _1} + 3{\lambda _2} = 1$ ..... (i)</p> <p>and $$\overrightarrow v \,.\,\widehat j = 7 \Rightarrow {\lambda _1} - 3{\lambda _2} = 7$$ ... (ii)</p> <p>from equation (i) and (ii)</p> <p>${\lambda _1} = 4$, ${\lambda _2} = - 1$</p> <p>$\therefore$ $\overrightarrow v = 2\widehat i + 7\widehat j + 7\widehat k$</p> <p>$\therefore$ $\overrightarrow v \,.\,(\widehat i + \widehat k) = 2 + 7$</p> <p>$= 9$</p>