Question

If $(\overrightarrow{a}+3\overrightarrow{b}).(7\overrightarrow{a}-5\overrightarrow{b})=0$ and $(\overrightarrow{a}-5\overrightarrow{b}).(7\overrightarrow{a}+3\overrightarrow{b})=0$, then angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

• A. ${60}^o$
• B. ${30}^o$
• C. ${90}^o$
• D. $Noneofthese$

Answer 1

Answer: (C)

${90}^o$

$(\overline{a}+3\overline{b}).(7\overline{a}-5\overline{b})=0$

$7a^2-5a\overline{b}+21\overline{a}\overline{b}-15b^2=0(\because a.b=b.a)$

$7a^2+16a\overline{b}-15b^2=0$ _______ (1)

$(\overline{a}-5\overline{b})(7\overline{a}+3\overline{b})=0$

$7a^2+3\overline{a}\overline{b}-35\overline{a}\overline{b}-15b^2=0$

$7a^2-32\overline{a}\overline{b}-15b^2=0$ ______ (2)

Subtracting, 48.a.b = 0

a.b = 0

Thus ,$\overline{a}$ is perpendicular to $\overline{b}$

$\therefore$ Angle = ${90}^{\circ }$