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Use the cross product to compute the area of triangles with given vertices

For each of the following sets of points A,B,C use the cross product to compute the area of the triangle with vertices A,B,C.

  1. A = (0,2,2), \quad B = (2,0,-1), \quad C = (3,4,0);
  2. A = (-2,3,1), \quad B = (1,-3,4), \quad C = (1,2,1);
  3. A = (0,0,0), \quad B = (0,1,1), \quad C = (1,0,1).

  1. The area of the triangle with vertices A,B,C is given by

        \begin{align*}  \text{Area} &= \frac{1}{2} \lVert (A - C) \times (B-C) \rVert \\  &= \frac{1}{2} \lVert (-3,-2,2) \times (-1,-4,-1) \rVert \\  &= \frac{1}{2} \lVert (10,-5,10) \rVert \\  &= \frac{\sqrt{225}}{2} \\  &= \frac{15}{2}. \end{align*}

  2. The area of the triangle with vertices A,B,C is given by

        \begin{align*}  \text{Area} &= \frac{1}{2} \lVert (A - C) \times (B-C) \rVert \\  &= \frac{1}{2} \lVert (-3,1,0) \times (0,-5,3) \rVert \\  &= \frac{1}{2} \lVert (3,9,15) \rVert \\  &= \frac{\sqrt{333}}{2} \\  &= \frac{3}{2} \sqrt{37}. \end{align*}

  3. The area of the triangle with vertices A,B,C is given by

        \begin{align*}  \text{Area} &= \frac{1}{2} \lVert (A - C) \times (B-C) \rVert \\  &= \frac{1}{2} \lVert (-1,0,-1) \times (-1,1,0) \rVert \\  &= \frac{1}{2} \lVert (1,1,-1) \rVert \\  &= \frac{\sqrt{3}}{2}. \end{align*}

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