QJEEM21Ma74ca: JEE Main 2021 Physics - Class 11 JEEM21Ma74ca Solution: If vectors and are collinear, then a possible unit vector…

Question

Verified Solution • 2025–26 Session moderate

If vectors

\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k

and

\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat k

are collinear, then a possible unit vector parallel to the vector

x\widehat i + y\widehat j + z\widehat k

is :

Accepted Answer

\overrightarrow {{a_2}} = \lambda \overrightarrow {{a_1}}

\widehat i + y\widehat j + z\widehat k = \lambda (x\widehat i - \widehat j + \widehat k)

1 = \lambda x,y = - \lambda ,z = \lambda

x\widehat i + y\widehat j + z\widehat k = {1 \over \lambda }\widehat i - \lambda \widehat j + \lambda \widehat k

Unit vector

= {{{1 \over \lambda }\widehat i - \lambda \widehat j + \lambda \widehat k} \over {\sqrt {{1 \over {{\lambda ^2}}} + {\lambda ^2} + {\lambda ^2}} }}

= {{\widehat i - {\lambda ^2}\widehat j + {\lambda ^2}\widehat k} \over {\sqrt {1 + 2{\lambda ^4}} }}

Let

{\lambda ^2} = 1

, possible unit vector

= {{\widehat i - \widehat j + \widehat k} \over {\sqrt 3 }}

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