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$\mathop {\,{\rm{A}}}\limits^ \to \,\, + \;\,\mathop {\rm{B}}\limits^ \to \,\, + \,\,\mathop {\rm{C}}\limits^ \to \, = \,\,\mathop 0\limits^ \to $ આપેલ છે. ત્રણ સદિશ પૈકી બે સદિશોનું મૂલ્ય સમાન છે. અને ત્રીજા સદિશનું મૂલ્ય $\sqrt 2 $ ગણું કે જે બે સમાન મૂલ્ય સિવાયનું છે. તો સદિશો વચ્ચેના ખૂણાઓ શું હશે ?
$30^°, 60^°,90^°$
$45^°,45^°,90^°$
$45^°,60^°,90^°$
$90^°,135^°,135^°$
Solution
$\begin{array}{l}
\mathop {\,{\rm{A}}}\limits^ \to \,\, + \;\,\mathop {\rm{B}}\limits^ \to \,\, + \,\,\mathop {\rm{C}}\limits^ \to \, = \,\,\mathop 0\limits^ \to \, \Rightarrow \,\,\mathop {\,{\rm{A}}}\limits^ \to \,\, + \;\,\mathop {\rm{B}}\limits^ \to \,\, = – \,\,\mathop {\rm{C}}\limits^ \to \\
{A^2}\,\, + \;\,{B^2}\,\, + \;\,2AB\cos {\theta _1}\,\, = \,\,{C^2}\,\,\left( {\,\,A\,\, = \,B} \right)\\
{B^2}\,\, + \;\,{B^2}\,\, + \;\,2AB\cos {\theta _1}\,\, = \,\,2{B^2}\,\,\left( {\,\,C\,\, = \,\,\sqrt 2 B} \right)\\
2AB\cos {\theta _1}\,\, = \,\,0\,\, \Rightarrow \,\,\cos {\theta _1}\,\, = \,\,0\,\, \Rightarrow \,\,{\theta _1}\,\, = \,\,90^\circ \\
\mathop {\rm{B}}\limits^ \to \,\, + \,\,\mathop {\rm{C}}\limits^ \to \,\, = \,\, – \mathop {\rm{A}}\limits^ \to \,\, \Rightarrow \,\,{B^2}\,\, + \;\,{C^2}\,\, + \;\,2BC\,\,\cos {\theta _2}\,\, = \,\,{A^2}\\
{B^2}\,\, + \;2{B^2}\,\, + \;\,2B\sqrt 2 \,B\cos {\theta _2}\,\, = \,\,{B^2}\\
2{B^2}\,\,\left( {1\, + \,\,\sqrt 2 \cos {\theta _2}} \right)\,\, = \,\,0\,\, \Rightarrow \,\,\cos {\theta _2}\,\, = \,\, – \frac{1}{{\sqrt 2 }}\,\, \Rightarrow \,\,{\theta _2}\,\, = \,\,135^\circ \\
\therefore \,\,\left( {90^\circ ,\,135^\circ ,135^\circ } \right)
\end{array}$ ખૂણાઓ છે