LinAlg4.1.14

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Prove that three points whose vectors are {\bar  {i}}+2{\bar  {j}}+3{\bar  {k}},-{\bar  {i}}-{\bar  {j}}+8{\bar  {k}},4{\bar  {i}}+4{\bar  {j}}+6{\bar  {k}}\, form an equilateral triangle.

Let A={\bar  {i}}+2{\bar  {j}}+3{\bar  {k}},B=-{\bar  {i}}-{\bar  {j}}+8{\bar  {k}},C=4{\bar  {i}}+4{\bar  {j}}+6{\bar  {k}}\,

Then {\bar  {AB}}=-2{\bar  {i}}-3{\bar  {j}}+5{\bar  {k}},{\bar  {BC}}=-3{\bar  {i}}+5{\bar  {j}}-2{\bar  {k}},{\bar  {CA}}=5{\bar  {i}}-2{\bar  {j}}-3{\bar  {k}}\,

Now |AB|={\sqrt  {4+9+25}}={\sqrt  {38}},|BC|={\sqrt  {9+25+4}}={\sqrt  {38}},|CA|={\sqrt  {24+9}}={\sqrt  {38}}\,

Thus |AB|=|BC|=|CA|\,.Hence the points form an equilateral triangle.


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