LinAlg4.1.15

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Show that the triangle ABC whose vertices are 7{\bar  {i}}+10{\bar  {k}},-{\bar  {i}}+6{\bar  {j}}+6{\bar  {k}},-4{\bar  {i}}+9{\bar  {j}}+6{\bar  {k}}\, is isoscles and right angled.

Given A=7{\bar  {i}}+10{\bar  {k}},B=-{\bar  {i}}+6{\bar  {j}}+6{\bar  {k}},C=-4{\bar  {i}}+9{\bar  {j}}+6{\bar  {k}}\,

{\bar  {AB}}=-{\bar  {i}}-{\bar  {j}}-4{\bar  {k}},{\bar  {BC}}=-3{\bar  {i}}+3{\bar  {j}},{\bar  {CA}}=4{\bar  {i}}-2{\bar  {j}}+4{\bar  {k}}\,

AB=|AB|={\sqrt  {1+1+16}}={\sqrt  {18}}=3{\sqrt  {2}},BC=|BC|={\sqrt  {9+9}}={\sqrt  {18}}=3{\sqrt  {2}},CA=|CA|={\sqrt  {16+4+16}}=6\,

Thus AB=BC=3{\sqrt  {2}}\, Triangle is isosceles and also AB^{{2}}+BC^{{2}}=18+18=36=CA^{{2}}\, which is right angled.


Hence the triangle ABC is right angled isosceles.


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