EXAMPLE 4 Find the area of the triangle with
vertices \(P(1, 4, 6), Q(-2, 5, -1),\)
and \(R(1, -1, 1)\).
SOLUTION In Example 3 we computed that \(\vec{PQ} \times \vec{PR} = \langle -40, -15, 15
\rangle\). The area of the parallelogram with adjacent sides PQ
and PR is the length of this cross product: \[
|\vec{PQ} \times \vec{PR}| = \sqrt{(-40)^2 + (-15)^2 + 15^2} =
\sqrt{1600 + 225 + 225} = \sqrt{2050} = 5\sqrt{82}
\] The area A of the triangle PQR is half the area of this
parallelogram, that is, \(\frac{5}{2}\sqrt{82}\).