QJEEM21Ma2794: JEE Main 2021 Physics - Class 11 JEEM21Ma2794 Solution: Let and . Let a vector be in the plane containing and . If is…

Question

Verified Solution • 2025–26 Session moderate

Let

\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k

and

\overrightarrow b = \widehat i + 2\widehat j - \widehat k

. Let a vector

\overrightarrow v

be in the plane containing

\overrightarrow a

and

\overrightarrow b

. If

\overrightarrow v

is perpendicular to the vector

3\widehat i + 2\widehat j - \widehat k

and its projection on

\overrightarrow a

is 19 units, then

{\left| {2\overrightarrow v } \right|^2}

is equal to _____________.

Accepted Answer

\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k

\overrightarrow b = \widehat i + 2\widehat j - \widehat k

\overrightarrow c = 3\widehat i + 2\widehat j - \widehat k

\overrightarrow v = x\overrightarrow a + y\overrightarrow b

\overrightarrow v \left( {3\widehat i + 2\widehat j - \widehat k} \right) = 0

\overrightarrow v .\widehat a = 19

\overrightarrow v = \lambda \overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } \right)

\overrightarrow v = \lambda \left[ {\left( {\overrightarrow c .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow c .\overrightarrow a } \right)\overrightarrow b } \right]

= \lambda [(3 + 4 + 1)\left( {2\widehat i - \widehat j + 2\widehat k} \right) - \left( {{{6 - 2 - 2} \over 2}} \right)\left( {\widehat i + 2\widehat j + \widehat k} \right)

= \lambda [16\widehat i - 8\widehat j + 16\widehat k - 2\widehat i - 4\widehat j + 2\widehat k]

\overrightarrow v = \lambda \left[ {14\widehat i - 12\widehat j + 18\widehat k} \right]

\lambda [14\widehat i - 12\widehat j + 18\widehat k].{{\left( {2\widehat i - \widehat j + 2\widehat k} \right)} \over {\sqrt {4 + 1 + 4} }} = 19

\lambda {{[28 + 12 + 36]} \over 3} = 19

\lambda \left( {{{76} \over 3}} \right) = 19

4\lambda = 3 \Rightarrow \lambda = {3 \over 4}

|2{v^2}| = {\left| {2 \times {3 \over 4}(14\widehat i - 12\widehat j + 18\widehat k)} \right|^2}

{9 \over 4} \times 4{\left( {7\widehat i - 6\widehat j + 9\widehat k} \right)^2}

= 9(49 + 36 + 81)

= 9(166)

= 1494

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