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  • Show that the points A , B ,C with position vectors 2i-j+k , i-3j-5k and 3i-4j-4k respectively are vertices of right-angled Triangle . Hence find Area of triangle.

Show that the points A , B ,C with position vectors 2i-j+k , i-3j-5k and 3i-4j-4k respectively are vertices of right-angled Triangle . Hence find Area of triangle.

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Solution:   Vectors ,

             vecOA=2hati-hatj+hatkhspace0.2cm,vecOB=hati-3hatj-5hatkhspace0.2cmandhspace0.2cmvecOC=3hati-4hatj-4hatk

         The igtriangleup hspace0.1cmABC is a right angled triangle and find the Area of igtriangleup hspace0.1cmABC

        Then ,

                vecAB=vecOB-vecOA=-hati-2hatj+6hatk

               vecBC=vecOC-vecOB=2hati-hatj+hatk

               vecCA=vecOA-vecOC=-hati+3hatj+5hatk

        By Pathagoras Theoram ,   left | vecAB ight |^2=left | arBC ight |^2+left | vecCA ight |^2

              Rightarrow                                  41=6+35

      Hence proved igtriangleup hspace0.1cmABC is a right-angled triangle at  C

     Rightarrow          Area of igtriangleup hspace0.1cmABC=(1/2)left | vecBC ight | imes left | vecCA ight |=(1/2) imes sqrt6 imes sqrt35=sqrt210/2

Posted by

Deependra Verma

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