JEE Main & Advanced Physics Vectors Sample Paper Topic Test- Vectors

If \[{{\bar{a}}_{1}}\] and \[{{\bar{a}}_{2}}\] are two non-collinear unit vectors and \[|{{\bar{a}}_{1}}+{{\bar{a}}_{2}}|=\sqrt{3},\] then the value of \[({{\bar{a}}_{1}}-{{\bar{a}}_{2}}).(2{{\bar{a}}_{1}}+{{\bar{a}}_{2}})\] is

A) 2

B) 3/2

C) ½

D) 1

Correct Answer: C

Solution :

\[|{{\overset{-}{\mathop{a}}\,}_{_{1}}}+{{\overset{-}{\mathop{a}}\,}_{_{2}}}|=\sqrt{a_{1}^{2}+a_{2}^{2}+2{{a}_{1}}{{a}_{2}}\cos \theta }=\sqrt{3}\]

\[\Rightarrow \cos \theta =\frac{1}{2}\]

\[({{\overset{-}{\mathop{a}}\,}_{1}}-{{\overset{-}{\mathop{a}}\,}_{2}})(2{{\overset{-}{\mathop{a}}\,}_{1}}+{{\overset{-}{\mathop{a}}\,}_{2}})=\]

\[2a_{1}^{2}-a_{2}^{2}-{{a}_{1}}{{a}_{2}}\cos \theta =\frac{1}{2}\]