91Ó°ÊÓ

To find a plane equation that passes through points \(P\), \(Q\), and \(R\), first determine two vectors that lie on the plane. Calculate the vectors \( \mathbf{PQ} \) and \( \mathbf{PR} \). For a vector between two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), use the formula:\[\mathbf{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\].\[\mathbf{PQ} = (4 - 3, \frac{2}{3} - \frac{1}{3}, -3 - (-5)) = (1, \frac{1}{3}, 2)\]\[\mathbf{PR} = (2 - 3, 0 - \frac{1}{3}, 1 - (-5)) = (-1, -\frac{1}{3}, 6)\]

The normal vector \(\mathbf{n}\) to the plane is perpendicular to both \(\mathbf{PQ}\) and \(\mathbf{PR}\). Find \(\mathbf{n}\) by computing the cross product \(\mathbf{PQ} \times \mathbf{PR}\). Use the determinant form to compute the cross product:\[\mathbf{n} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \1 & \frac{1}{3} & 2 \-1 & -\frac{1}{3} & 6\end{vmatrix}\]Now, calculate the determinant:\[\mathbf{n} = \left| \begin{array}{ll} \frac{1}{3} & 2 \ -\frac{1}{3} & 6 \end{array} \right| \mathbf{i} - \left| \begin{array}{ll} 1 & 2 \ -1 & 6 \end{array} \right| \mathbf{j} + \left| \begin{array}{ll} 1 & \frac{1}{3} \ -1 & -\frac{1}{3} \end{array} \right| \mathbf{k}\]Evaluating these determinants, we get:\[\mathbf{n} = \left( \left(\frac{1}{3} \cdot 6 - \frac{1}{3} \cdot 2\right), -\left( 1 \cdot 6 - (-1) \cdot 2 \right), \left( 1 \cdot (-\frac{1}{3}) - (-1) \cdot \frac{1}{3} \right) \right)\]\[\mathbf{n} = (2, -8, 0)\]

The equation of a plane with normal vector \( \mathbf{n} = (A, B, C) \) passing through a point \( (x_0, y_0, z_0) \) can be written as:\[A(x - x_0) + B(y - y_0) + C(z - z_0) = 0\]Substitute \(A = 2\), \(B = -8\), \(C = 0\), and the point \(P(3, \frac{1}{3}, -5)\):\[2(x - 3) - 8\left(y - \frac{1}{3}\right) + 0(z + 5) = 0\]Simplifying this gives:\[2x - 6 - 8y + \frac{8}{3} = 0\]Combine like terms and rearrange:\[2x - 8y - \frac{10}{3} = 0\]

To simplify the equation \(2x - 8y - \frac{10}{3} = 0\), multiply every term by 3 to eliminate the fraction:\[6x - 24y - 10 = 0\]This is the equation of the plane in a simplified form.