LinAlg4.2.13

From Exampleproblems

Find a unit vector perpendiculars to the plane ABC where $A=(3,-1,2),B(1,-1,-3),C(4,-3,1)\,$

Given $A=(3,-1,2),B(1,-1,-3),C(4,-3,1)\,$

$\bar{AB}=-2\bar{i}-5\bar{k},\bar{AC}=\bar{i}-2\bar{j}-\bar{k}\,$

Now $\bar{AB} \times \bar{AC}= \begin{vmatrix} \bar{i} & \bar{j} & \bar{k} \\ -2 & 0 & -5 \\ 1 & -2 & -1 \end{vmatrix}\,$ which is equal to

$\bar{i}(0-10)-\bar{j}(2+5)+\bar{k}(4-0)=-10\bar{i}-7\bar{j}+4\bar{k}\,$

Therefore,a unit vector perpendicular to the plane ABC = $\pm \frac{\bar{AB}\times \bar{AC}}{|AB \times AC|}=\pm\frac{-10\bar{i}-7\bar{j}+4\bar{k}}{\sqrt{(-10)^2+(-7)^2+4^2}=\pm\frac{1}\sqrt{165}}(-10\bar{i}-7\bar{j}+4\bar{k})\,$

Main Page:Linear Algebra

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