To find the equation of the plane passing through the points \( P(1, 1, 1) \), \( Q(3, -1, 2) \), and \( R(-3, 5, -4) \), we can follow these steps:
### Step 1: Identify the points
We have the points:
- \( P(1, 1, 1) \) which gives us \( (x_1, y_1, z_1) = (1, 1, 1) \)
- \( Q(3, -1, 2) \) which gives us \( (x_2, y_2, z_2) = (3, -1, 2) \)
- \( R(-3, 5, -4) \) which gives us \( (x_3, y_3, z_3) = (-3, 5, -4) \)
### Step 2: Set up the equation of the plane
The equation of a plane through three non-collinear points can be expressed using the determinant:
\[
\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}
\]
This can be written as:
\[
\begin{vmatrix}
x - x_1 & y - y_1 & z - z_1 \\
x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1
\end{vmatrix} = 0
\]
### Step 3: Substitute the values
Substituting the coordinates of points \( P \), \( Q \), and \( R \):
\[
\begin{vmatrix}
x - 1 & y - 1 & z - 1 \\
3 - 1 & -1 - 1 & 2 - 1 \\
-3 - 1 & 5 - 1 & -4 - 1
\end{vmatrix} = 0
\]
This simplifies to:
\[
\begin{vmatrix}
x - 1 & y - 1 & z - 1 \\
2 & -2 & 1 \\
-4 & 4 & -5
\end{vmatrix} = 0
\]
### Step 4: Calculate the determinant
Now, we will calculate the determinant:
\[
= (x - 1) \begin{vmatrix}
-2 & 1 \\
4 & -5
\end{vmatrix} - (y - 1) \begin{vmatrix}
2 & 1 \\
-4 & -5
\end{vmatrix} + (z - 1) \begin{vmatrix}
2 & -2 \\
-4 & 4
\end{vmatrix}
\]
Calculating each of these 2x2 determinants:
1. \(\begin{vmatrix}
-2 & 1 \\
4 & -5
\end{vmatrix} = (-2)(-5) - (1)(4) = 10 - 4 = 6\)
2. \(\begin{vmatrix}
2 & 1 \\
-4 & -5
\end{vmatrix} = (2)(-5) - (1)(-4) = -10 + 4 = -6\)
3. \(\begin{vmatrix}
2 & -2 \\
-4 & 4
\end{vmatrix} = (2)(4) - (-2)(-4) = 8 - 8 = 0\)
Putting it all together:
\[
(x - 1)(6) - (y - 1)(-6) + (z - 1)(0) = 0
\]
This simplifies to:
\[
6(x - 1) + 6(y - 1) = 0
\]
### Step 5: Simplify the equation
Expanding and simplifying:
\[
6x - 6 + 6y - 6 = 0 \implies 6x + 6y - 12 = 0
\]
Dividing the entire equation by 6:
\[
x + y - 2 = 0
\]
### Final Equation
Thus, the equation of the plane passing through the points \( P \), \( Q \), and \( R \) is:
\[
x + y = 2
\]
