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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