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 :
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Official solution
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