InterviewSolution
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InterviewSolution
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सिद्ध कीजिये कि `DeltaABC` समबाहु होगा, यदि A, B तथा C के स्थिति-सदिश क्रमशः `hati+2hatj+3hatk,-hati-hatj+8hatk` तथा `-4hati+4hatj+6hatk` है। |
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Answer» माना मूलबिन्दु O के सापेक्ष बिन्दुओं A, B तथा C के स्थिति-सदिश क्रमशः `hati+2hatj+3hatk,-hati-hatj+8hatk` तथा `-4hati+4hatj+6hatk` है। `vec(OA)=hati+2hatj+3hatk, vec(OB)=-hati-hatj+8hatk` तथा `vec(OC)=-4hati+4hatj+6hatk` `vec(AB)=vec(OB)-vec(OA)` `=(-hati-hatj+8hatk)-(hati+2hatj+3hatk)` `=(-hati-hati)+(-hatj-2hatj)+(8hatk-3hatk)` `=-2hati-3hatj+5hatk` `|vec(AB)|=|-2hati-3hatj+5hatk|=sqrt((-2)^(2)+(-3)^(2)+(5)^(2))` `=sqrt(4+9+25)=sqrt(38)` `vec(BC)=vec(OC)-vec(OB)` `=(-4hati+4hatj+6hatk)-(-hati-hatj+8hatk)` `=(-4hati+hati)+(4hatj+hatj)+(6hatk-8hatk)` `=-3hati+5hatj-2hatk` `|vec(BC)|=|-3hati+5hatj-2hatk|` `=sqrt((-3)^(2)+(5)^(2)+(-2)^(2))` `=sqrt(9+25+4)=sqrt(38)` `vec(AC)=vec(OC)-vec(OA)` `=(-4hati+4hatj+6hatk)-(hati+2hatj+3hatk)` `=(-4hati-hati)+(4hatj-2hatj)+(6hatk-3hatk)` `=-5hati+2hatj+3hatk` `|vec(AC)|=|-5hati+2hatj+3hatk|` `=sqrt((-5)^(2)+(2)^(2)+(3)^(2))=sqrt(25+4+9)=sqrt(38)` `because |vec(BC)|=|vec(AC)|=|vec(AB)|` अतः `DeltaABC` समबाहु त्रिभुज है। |
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