LinAlg4.2.12

From Exampleproblems

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.

Main Page:Linear Algebra

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