Find the area of the parallelogram with vertices k(1, 1, 1), l(1, 3, 2), m(5, 8, 2), and n(5, 6, 1).

The area can be found as the magnitude of the cross product of the vectors representing two adjacent sides.
  area = || kl × kn || = || (0, 2, 1) × (4, 5, 0) ||
    = || (-5, 4, -8) || = √105

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For the purpose here, kl = l - k = (1, 3, 2) - (1, 1, 1) = (0, 2, 1)
and nk = n - k = (5, 6, 1) - (1, 1, 1) = (4, 5, 0)

The cross product can be calculated using a suitable calculator. By hand, it is the determinant

where i, j, k are unit vectors in the x, y, z directions.

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