To find the equation of a plane passing through three points, $P(-4, -1, -2)$, $Q(5, 2, 8)$, and $R(-1, 0, 5)$, we can proceed with the following steps:

1. Calculate Two Direction Vectors:

   • Vector $\vec{PQ}$ from $P$ to $Q$: $\vec{PQ} = Q - P = (5 - (-4), 2 - (-1), 8 - (-2)) = (9, 3, 10)$
   • Vector $\vec{PR}$ from $P$ to $R$: $\vec{PR} = R - P = (-1 - (-4), 0 - (-1), 5 - (-2)) = (3, 1, 7)$

2. Find the Cross Product $\vec{PQ} \times \vec{PR}$: The cross product of $\vec{PQ}$ and $\vec{PR}$ will give us a vector that is perpendicular to the plane, which we can use as the normal vector $\vec{n} = (a, b, c)$.

   $\vec{PQ} \times \vec{PR} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 9 & 3 & 10 \\ 3 & 1 & 7 \end{vmatrix}$ Expanding this determinant: $\vec{PQ} \times \vec{PR} = \hat{i}(3 \cdot 7 - 10 \cdot 1) - \hat{j}(9 \cdot 7 - 10 \cdot 3) + \hat{k}(9 \cdot 1 - 3 \cdot 3)$ $= \hat{i}(21 - 10) - \hat{j}(63 - 30) + \hat{k}(9 - 9)$ $= \hat{i}(11) - \hat{j}(33) + \hat{k}(0)$ $= (11, -33, 0)$

   So, the normal vector $\vec{n} = (11, -33, 0)$.

3. Form the Equation of the Plane: Using the point-normal form of the plane, $a(x - x_1) + b(y - y_1) + c(z - z_1) = 0$, where $(x_1, y_1, z_1)$ is a point on the plane (let’s use $P(-4, -1, -2)$) and $\vec{n} = (11, -33, 0)$:

   $11(x + 4) - 33(y + 1) + 0(z + 2) = 0$ Expanding this: $11x + 44 - 33y - 33 = 0$ $11x - 33y + 11 = 0$

4. Simplify the Equation: Divide through by 11 to simplify: $x - 3y + 1 = 0$

Thus, the equation of the plane passing through points $P$, $Q$, and $R$ is: $x - 3y + 1 = 0$

Would you like a detailed explanation of each step or have any questions about the process?

Here are five questions to expand your understanding:

1. How can the cross product of two vectors help in determining the equation of a plane?
2. What would change if we used a different point on the plane in the point-normal form?
3. Why is it necessary to simplify the equation after forming it initially?
4. How would you confirm if a fourth point lies on the same plane?
5. Can this approach be extended to higher-dimensional spaces?

Tip: Always ensure to check the result by plugging back at least one point into the final equation to verify its correctness.