$\therefore \,\,\mathop {BC}\limits^ \to \,\, = \,\,\mathop {AC}\limits^ \to \, - \,\mathop {AB}\limits^ \to \,\, = \,\,5\overline i \, - \,2\overline j \, + \,4\overline k \, - \,3\overline i \, - \,4\overline k \,\, = \,\,2\overline i \, - \,2\overline j \,\,,\,\,\,$
$AB\,\, = \,\,\sqrt {9\, + \,16} \,\, = \,\,5,\,$
$BC\,\, = \,\,\sqrt {4\, + \,4} \,\, = \,\,2\sqrt 2 ,\,$
$AC\,\, = \,\,\sqrt {25\, + \,4\, + \,16} \,\, = \,\,3\sqrt 5 $
$\therefore \,\,BD\,\, = \,\,\frac{1}{2}\,$ અને $\,BC\,\, = \,\,\sqrt 2 $
હવે એપોલોનીયસના પ્રમેય મુજબ
$A{B^2}\, + \,A{C^2}\,\, = \,\,\left( {A{D^2}\, + \,B{D^2}} \right)$
$\therefore \,\,25\, + \,45\,\, = \,\,2\,(A{D^2}\, + \,2)\,\,$
$\,\therefore \,\,35\,\, = \,\,A{D^2}\, + \,2\,\,\,$
$\therefore \,\,A{D^2}\,\, = \,\,33\,\,\,$
$\therefore \,\,AD\,\, = \,\,\sqrt {33} $
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