Question

If $|\overline{a}|=1,|\overline{b}|=2,|\overline{a}-\overline{b}{|}^2+|\overline{a}+2\overline{b}{|}^2=20,$ then $(\overline{a},\overline{b})=$

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. $\frac{2\pi }{3}$

Answer 1

Answer: (D)

$\frac{2\pi }{3}$

Given

$|\overline{a}|=1,|\overline{b}|=2,|\overline{a}-\overline{b}{|}^2+|\overline{a}+2\overline{b}{|}^2=20$

$|\overline{a}{|}^2+|\overline{b}{|}^2-2|\overline{a}||\overline{b}|\cos (\overline{a},\overline{b})+|\overline{a}{|}^2+4|\overline{b}{|}^2+4|\overline{a}||\overline{b}|\cos (\overline{a},\overline{b})=20$

$\Rightarrow 2|\overline{a}{|}^2+5|\overline{b}{|}^2+2|\overline{a}||\overline{b}|\cos (\overline{a},\overline{b})=20$

$\Rightarrow 2(1{)}^2+5(2{)}^2+2\times 1\times 2\times \cos (\overline{a},\overline{b})=20$

$\Rightarrow 22+4\cos (\overline{a},\overline{b})=20$

$\cos (\overline{a},\overline{b})=\frac{-1}{2}$

$\Rightarrow \pi -(\overline{a},\overline{b})=\frac{\pi }{3}$

$\Rightarrow (\overline{a},\overline{b})=(\pi -\frac{\pi }{3})=\frac{2\pi }{3}$