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# The vertices A, B, and C of a triangle in $\mathbb{R}^{3}$ are given. Find the lengths of the sides of the triangle and determine whether it is a right triangle, an isosceles triangle, both, or neither.$A(1,1,1), B(3,3,2), C(3,-3,5)$

Solution

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The given vectors are $A(1, 1, 1), B(3,3,2), C(3,-3,5)$. The component form of AB is

$\begin{gather} AB=(3,3,2)-(1, 1, 1)=(2,2,1) \end{gather}$

Now the distance is

$\begin{gather*} ||AB||=\sqrt{\left|{2}\right|^{2} + \left|{2}\right|^{2}+ \left|{1}\right|^{2}}=\sqrt{4+4+1} =\sqrt{9}=3 \end{gather*}$

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