LinAlg4.2.12

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IF (1,2,3),(2,5,-1),(-1,1,2)\, are the vertices of a triangle, find its area.

Let the points be A=(1,2,3),B=(2,5,-1),C=(-1,1,2)\,

Then {\bar  {AB}}={\bar  {i}}+3{\bar  {j}}-4{\bar  {k}},{\bar  {AC}}=-2{\bar  {i}}-{\bar  {j}}-{\bar  {k}}\,

Now {\bar  {AB}}\times {\bar  {AC}}={\begin{vmatrix}{\bar  {i}}&{\bar  {j}}&{\bar  {k}}\\1&3&-4\\-2&-1&-1\end{vmatrix}}\,

which is equal to {\bar  {i}}(-3-4)-{\bar  {j}}(-1-8)+{\bar  {k}}(-1+6)=-7{\bar  {i}}+9{\bar  {j}}+5{\bar  {k}}\,

Therefore,area of the triangle ABC={\frac  {1}{2}}|AB\times AC|={\frac  {1}{2}}|-7{\bar  {i}}+9{\bar  {j}}+5{\bar  {k}}|=frac{1}{2}{\sqrt  {(-7)^{2}+9^{2}+5^{2}}}={\frac  {1}{2}}({\sqrt  {49+81+25}}={\frac  {1}{2}}{\sqrt  {155}}\, square units.


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