QJEEM22Ma08d1: JEE Main 2022 Mathematics - Class 12 JEEM22Ma08d1 Solution: Let , and be three given vectors. Let be a vector in the plane…

Let $\overrightarrow v = {\lambda _1}\overrightarrow a + {\lambda _2}\overrightarrow b$ , where ${\lambda _1},\,{\lambda _2} \in R$ .

$= ({\lambda _1} + 2{\lambda _2})\widehat i + ({\lambda _1} - 3{\lambda _2})\widehat j + (2{\lambda _1} + {\lambda _2})\widehat k$

$\because$ Projection of $\overrightarrow v$ on $\overrightarrow c$ is ${2 \over {\sqrt 3 }}$

$\therefore$ ${{{\lambda _1} + 2{\lambda _2} - {\lambda _1} + 3{\lambda _2} + 2{\lambda _1} + {\lambda _2}} \over {\sqrt 3 }} = {2 \over {\sqrt 3 }}$

$\therefore$ ${\lambda _1} + 3{\lambda _2} = 1$ ..... (i)

and $\overrightarrow v \,.\,\widehat j = 7 \Rightarrow {\lambda _1} - 3{\lambda _2} = 7$ ... (ii)

from equation (i) and (ii)

${\lambda _1} = 4$ , ${\lambda _2} = - 1$

$\therefore$ $\overrightarrow v = 2\widehat i + 7\widehat j + 7\widehat k$

$\therefore$ $\overrightarrow v \,.\,(\widehat i + \widehat k) = 2 + 7$

$= 9$