Q. If ${\overrightarrow{A}}_{1}$ and ${\overrightarrow{A}}_{2}$ are two non-collinear unit vectors and if $\left| {\overrightarrow{A}}_{1} + {\overrightarrow{A}}_{2} \right| = \sqrt{3}$, then the value of (${\overrightarrow{A}}_{1}\mathbf{-}{\overrightarrow{A}}_{2}$)$\bullet (2{\overrightarrow{A}}_{1} + {\overrightarrow{A}}_{2})$ is
A) 1
B) $1/2$
C) 3/2
D) 2
Explanation
(b) Here, $A_{1} = A_{2} = 1$
and $A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}cos\theta = \left( \sqrt{3} \right)^{2} = 3$
or 1+1+2$\times 1 \times 1 \times$cos$\theta = 3$ or$\ cos\theta = \frac{1}{2}$
Now, $({\overrightarrow{A}}_{1} - {\overrightarrow{A}}_{2}) \bullet ({2\overrightarrow{A}}_{1} + {\overrightarrow{A}}_{2})$
$= 2A_{1}^{2} - A_{2}^{2} - A_{1}A_{2}\ cos\theta$
$= 2 \times 1^{2} - 1^{2} - 1 \times 1 \times \frac{1}{2} = \frac{1}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.