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Find a point satisfying conditions if given three vertices of a parallelogram

by RoRi

Let the three points

    \[ A = (1,0,1), \qquad B = (-1,1,1), \qquad C = (2,-1,2) \]

be three vertices of a parallelogram.

  1. Find the points D which are permissible fourth points of the parallelogram.
  2. Find the area of the triangle ABC.
  1. There are three possible points,

        \begin{align*} D &= B+C-A = (1,0,3) - (1,0,1) = (0,0,2) \\ D &= A+C-B = (3,-1,3) - (-1,1,1) = (4,-2,2) \\ D &= A+B-C = (0,1,2) - (2,-1,2) = (-2,2,0). \end{align*}

  2. The area of the triangle ABC is given by

        \[ \text{Area} = \frac{1}{2} \lVert A \times B \rVert = \frac{1}{2} \lVert (-1,-2,1) \rVert = \frac{\sqrt{6}}{2}. \]

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