$ \Rightarrow \,\,6{a^2} + \;3\,\vec a \,\,.\,\,\vec b \, - \,\,10\vec a \,.\,\,\vec b \,\, - 5{b^2}\,\, = \,\,0\,$
$ \Rightarrow \,\,6{a^2}\,\, - \,\,5{b^2}\,\, - \,\,7\vec a \,\,.\,\,\vec b \,\, = \,\,0\,......\left( i \right)$
અને $\left( {\vec a \, + \,\,4\vec b } \right)\,\,.\,\,\left( { - \vec a \, + \,\,\vec b } \right)\,\, = \,\,0$
$ \Rightarrow \,\, - {a^2}\,\, + \;_a^ \to \,\,.\,\,_b^ \to \,\, - \,\,4_a^ \to .\,\,_b^ \to \,\, + \;\,4{b^2}\,\, = \,\,0$
$\therefore \,\,{a^2}\,\, - \,\,4{b^2}\,\, + \,\,3_a^ \to \,\,.\,\,_b^ \to \,\,=\,\,0\,\,......\left( {ii} \right)$
સમીકરણ $(i)$ અને $(ii)$ માંથી $\vec a .\,\vec b $ નો લોપ કરતાં
$\,5a\,\, = \,\,\sqrt {43} b\,\,\,......\left( {iii} \right)$
સમીકરણ $(ii)$ પરથી
$\,\vec a .\,\vec b \,\, = \,\,\frac{{4{b^2}\,\, - \,\,{a^2}}}{3}\,\, = \,\,\frac{{4{b^2}\,\, - \,\,\frac{{43{b^2}}}{{25}}}}{3}\,\, = \,\,\frac{{57{b^2}}}{{75}}\,\, = \,\,\frac{{19{b^2}}}{{25}}$
જો $\vec a \,$ અને $\vec b $ વચ્ચેનો ખૂણો $\alpha $ હોય .
$\therefore \,\,\cos \,\,\alpha \, = \,\,\frac{{\vec a .\,\vec b }}{{|\vec a |\,\,|\vec b |}}\,\, = \,\,\frac{{\frac{{19{b^2}}}{{25}}}}{{\frac{{\sqrt {43b} }}{5}\,\, \times \,\,b}}\,\, = \,\,\frac{{19}}{{5\,\,\sqrt {43} }}$
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