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$P\,\, = \,\,{\rm{Q}}\,\, = \,\,{\rm{R}}$ જો $\mathop {\,{\rm{P}}}\limits^ \to \,\, + \;\,\mathop {\rm{Q}}\limits^ \to \,\, = \,\,\mathop {\rm{R}}\limits^ \to \,$ હોય તથા $\mathop {\rm{P}}\limits^ \to $ અને $\mathop {\rm{R}}\limits^ \to $ વચ્ચેનો ખૂણો ${\theta _1}$ છે. જો $\mathop {\rm{P}}\limits^ \to \,\, + \;\,\mathop {\rm{Q}}\limits^ \to \,\, + \,\,\mathop {\rm{R}}\limits^ \to \,\, = \,\,\mathop {\rm{0}}\limits^ \to $ હોય તો $\mathop {\rm{P}}\limits^ \to $ અને $\mathop {\rm{R}}\limits^ \to $ વચ્ચેનો ખૂણો ${\theta _2}$ છે. ${\theta _1}$ અને ${\theta _2}$ વચ્ચેનો સંબંધ શું કહે ?
${\theta _1} ={\theta _2}$
${\theta _1} ={\theta _2}/2 $
${\theta _1}={2\theta _2}$
ઉપરોક્ત એક પણ નહિ
Solution
$P\,\, = \,\,Q\,\, = \,\,R$
$\mathop {\,{\rm{P}}}\limits^ \to \,\, + \,\,\mathop {\,{\rm{Q}}}\limits^ \to \, = \,\,\mathop {\,{\rm{R}}}\limits^ \to \,\, \Rightarrow \,\mathop {\,{\rm{P}}}\limits^ \to \,\, – \,\,\mathop {\,{\rm{R}}}\limits^ \to \,\, = \,\, – \mathop {\,{\rm{Q}}}\limits^ \to $
${P^2}\,\, + \,\,{R^2}\,\, – \,\,2PR\cos {\theta _1}\,\, = \,\,{Q^2}\, \Rightarrow \,\,2\cos {\theta _1}\,\, = \,\,1\,\, \Rightarrow \,\,{\theta _1}\,\, = \,\,60^\circ $
હવે $\mathop {\,{\rm{P}}}\limits^ \to \,\, + \,\,\mathop {\,{\rm{Q}}}\limits^ \to \, + \mathop {\,{\rm{R}}}\limits^ \to \,\, = \,\,0\,\, \Rightarrow \,\,\mathop {\,{\rm{P}}}\limits^ \to \,\, + \,\mathop {\,{\rm{R}}}\limits^ \to \, = \,\, – \,\mathop {\,{\rm{Q}}}\limits^ \to \,$
$ \Rightarrow {P^2}\,\, + \;\,{R^2}\,\, + \,\,2PR\cos {\theta _2}\,\, = \,\,{Q^2}$
$ \Rightarrow \,\,\cos {\theta _2}\,\, = \,\, – \frac{1}{2}\,\, \Rightarrow \,\,{\theta _2}\,\, = \,\,120^\circ $ $ \Rightarrow \,\,{\theta _2}\,\, = \,\,2{\theta _1}\,\, \Rightarrow \,\,{\theta _2}\,\, = \,\frac{{{\theta _2}}}{2}$
