Question

If $(\overline{a}+3\overline{b}).(7\overline{a}-5\overline{b})=0$ and $(\overline{a}-4\overline{b}).(7\overline{a}-2\overline{b})=0$ then angle between $\overline{a}$ and $\overline{b}$ is

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

Answer 1

The correct option is C ${90}^o$

We have, $a+3b\perp 7a-5b$

$\Rightarrow (a+3b)\cdot (7a-5b)=0$

$\Rightarrow 7a^2-5a\cdot b+21a\cdot b-15b^2=0$

$\Rightarrow 7a^2+16a\cdot b-15b^2=0$----------------------(1)

Also, $a-5b\perp 7a+3b$

$\Rightarrow (a-5b)\cdot (7a+3b)=0$

$\Rightarrow 7a^2-35a\cdot b+3a\cdot b-15b^2=0$

$\Rightarrow 7a^2-32a\cdot b-15b^2=0$-------------------------(2)

Substracting Equation (2) from (1), we get

$48a\cdot b=0$

$a\cdot b=0$

$\therefore a\perp b$

$\therefore$ Angle between a & b is ${90}^{\circ }$.