Question

If $a_1$ and $a_2$ are two non-collinear unit vectors and if $|a_1 + a_2| = \sqrt3$ ,then the value of $(a_1 - a).(2a_1 + a_2)$ is

• A. 2
• B. $\frac{3}{2}$
• C. $\frac{1}{2}$
• D. 1

Answer 1

Answer: (C)

$\frac{1}{2}$

Answer 2

Since, $a_1$ and $a_2$ are non-collinear $\therefore a_1 = a_2 = 1$ and $|a_1 + a_2| = \sqrt3$ $\Rightarrow a_1^2 + a_2^2 + 2 a_1 a_2 \, \cos \, \theta = (\sqrt 3)^2$ $\Rightarrow 1 + 1 + 2\, \cos \, \theta = 3 \, \Rightarrow \, \cos \, \theta = \frac{1}{2}$ Now $(a_1 + a_2). (2 a_1 + a_2)$ $20mm = 2 a_1^2 - a_2^2 - a_1 a_2 \, \cos \, \theta = 2 -1 - \frac{1}{2} = \frac{1}{2}$