Exercise 46

  1. Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance \(d\) from P to the plane is \[ d = \frac{|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|}{|\mathbf{a} \times \mathbf{b}|} \] where \(\mathbf{a} = \vec{QR}, \mathbf{b} = \vec{QS},\) and \(\mathbf{c} = \vec{QP}\).
  2. Use the formula in part (a) to find the distance from the point P(2, 1, 4) to the plane through the points Q(1, 0, 0), R(0, 2, 0), and S(0, 0, 3).