QJEEM20Ma20c6: JEE Main 2020 Physics - Class 11 JEEM20Ma20c6 Solution: A vector lies in the plane of the vectors, and . If bisects the …

Question

Verified Solution • 2025–26 Session moderate

A vector

\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)

lies in the plane of the vectors,

\overrightarrow b = \widehat i + \widehat j

and

\overrightarrow c = \widehat i - \widehat j + 4\widehat k

. If

\overrightarrow a

bisects the angle between

\overrightarrow b

and

\overrightarrow c

, then:

Accepted Answer

Angle bisector

\overrightarrow a = \lambda \left( {\widehat b + \widehat c} \right)

=

\lambda \left( {{{\widehat i + \widehat j} \over {\sqrt 2 }} + {{\widehat i - \widehat j + 4\widehat k} \over {3\sqrt 2 }}} \right)

\Rightarrow

\overrightarrow a = {\lambda \over {3\sqrt 2 }}\left( {4\widehat i + 2\widehat j + 4\widehat k} \right)

comparing with

\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k

{{2\lambda } \over {3\sqrt 2 }}

= 2

\Rightarrow

\lambda

=

{3\sqrt 2 }

\therefore

\overrightarrow a = \left( {4\widehat i + 2\widehat j + 4\widehat k} \right)

Then

\overrightarrow a .\widehat k - 4

= 4 - 4 = 0

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