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. none of these

Answer 1

The correct option is A ${60}^o$

$\begin{array}{c}\left(a+3b\right).\left(7a+5b\right)=0\\ 7a^2-5a.b+21a.b-15b^2=0\\ 7a^2+16ab-15b^2=\mathrm{0..............}\left(1\right)\\ similarly\\ \left(a-4b\right)\left(7a-2b\right)=0\\ 7a^2-2ab-28ab+8b^2=0\\ 7a^2-30ab+8b^2=\mathrm{0.................}\left(2\right)\end{array}$

Now subtracting equation $\left(1\right)and\left(2\right)$

$\begin{array}{c}\left(7a^2+16ab-15b^2=0\right)-\left(7a^2-30ab+8b^2=0\right)\\ 46ab-23b^2=0\end{array}$

$b$ is unit vector

$\begin{array}{c}\therefore \left|b\right|=1\\ b^2=1\end{array}$

therefore

$\begin{array}{c}46ab-23\times 1=0\\ ab=\frac{1}{2}\\ \left|a\right|\left|b\right|\cos \theta =\frac{1}{2}\\ \cos \theta =\cos \frac{\pi }{3}\end{array}$

angle between $a$and $b$ is $\frac{\pi }{3}$