Question

Consine of an angle between the vectors $(\overrightarrow{a}+\overrightarrow{b})$ and $(\overrightarrow{a}-\overrightarrow{b})$ if $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=1$ and $\angle ={60}^{\circ }.$

• A. $\sqrt{3/7}$
• B. $9/\sqrt{21}$
• C. $3/\sqrt{7}$
• D. none

Answer 1

The correct option is A $\sqrt{3/7}$

$\cos \theta =\frac{(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b})}{|\overrightarrow{a}+\overrightarrow{b}||\overrightarrow{a}-\overrightarrow{b}|}$

$=\frac{a^2-\overrightarrow{a}\overrightarrow{b}+\overrightarrow{a}\overrightarrow{b}-b^2}{\sqrt{a^2+b^2+2ab\cos \alpha }\sqrt{a^2+b^2-2ab\cos \alpha }}$

$=\frac{4-1}{\sqrt{4+1+4cos{60}^0}\sqrt{2+1-4cos{60}^0}}$

$=\frac{3}{\sqrt{7}\sqrt{3}}=\sqrt{\frac{3}{7}}$