Two unit vectors ' $\hat{a}_{1}$ 'and ' $\hat{a}_{2}$ ' are inclined to each other at an angle ' $\theta$ '.…

Two unit vectors ' $\hat{a}_{1}$ 'and ' $\hat{a}_{2}$ ' are inclined to each other at an angle ' $\theta$ '. If $\left|\hat{a}_{1}-\hat{a}_{2}\right|=\sqrt{3}$, then the value of $\left(\hat{a}_{1}-\hat{a}_{2}\right) \cdot\left(2 \hat{a}_{1}-\hat{a}_{2}\right)$ is
  1. -$\frac{1}{2}$
  2. 2
  3. 1
  4. $\frac{3}{2}$

Solution

$\left|\hat{a}_{1}-\hat{a}_{2}\right|=\sqrt{3} \rightarrow(1)$ $\left|\hat{a}_{1}\right|=\left|\hat{a}_{2}\right|=1$ $\hat{a}_{1}^{2}+\hat{a}_{2}^{2}-2 \hat{a}_{1} \hat{a}_{2} \cos \theta=3 \quad$ squaring (1) $1+1-2 \cos \theta=3 \quad=\left(\hat{a}_{1}-\hat{a}_{2}\right) \cdot\left(2 \hat{a}_{1}-\hat{a}_{2}\right)$ $\cos \theta=-1 / 2$ .

Asked in: MHT CET 2020 (14 Oct Shift 2)

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