To find the equation of the plane passing through the points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), and the origin \((0, 0, 0)\), we can use the determinant method. Here’s a step-by-step solution:
### Step 1: Set up the determinant
The equation of a plane passing through three points can be represented using a determinant. The points we have are:
- Point A (the origin): \((0, 0, 0)\)
- Point B: \((x_1, y_1, z_1)\)
- Point C: \((x_2, y_2, z_2)\)
The equation of the plane can be expressed in the form of a determinant:
\[
\begin{vmatrix}
x & y & z \\
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2
\end{vmatrix} = 0
\]
### Step 2: Expand the determinant
The determinant can be expanded as follows:
\[
x \begin{vmatrix}
y_1 & z_1 \\
y_2 & z_2
\end{vmatrix} - y \begin{vmatrix}
x_1 & z_1 \\
x_2 & z_2
\end{vmatrix} + z \begin{vmatrix}
x_1 & y_1 \\
x_2 & y_2
\end{vmatrix} = 0
\]
### Step 3: Calculate the 2x2 determinants
Now we calculate the 2x2 determinants:
1. For the first determinant:
\[
\begin{vmatrix}
y_1 & z_1 \\
y_2 & z_2
\end{vmatrix} = y_1 z_2 - y_2 z_1
\]
2. For the second determinant:
\[
\begin{vmatrix}
x_1 & z_1 \\
x_2 & z_2
\end{vmatrix} = x_1 z_2 - x_2 z_1
\]
3. For the third determinant:
\[
\begin{vmatrix}
x_1 & y_1 \\
x_2 & y_2
\end{vmatrix} = x_1 y_2 - x_2 y_1
\]
### Step 4: Substitute back into the equation
Substituting these values back into the equation gives us:
\[
x(y_1 z_2 - y_2 z_1) - y(x_1 z_2 - x_2 z_1) + z(x_1 y_2 - x_2 y_1) = 0
\]
### Step 5: Rearranging the equation
This can be rearranged to give the equation of the plane:
\[
x(y_1 z_2 - y_2 z_1) - y(x_1 z_2 - x_2 z_1) + z(x_1 y_2 - x_2 y_1) = 0
\]
### Final Answer
Thus, the equation of the plane passing through the points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), and the origin is:
\[
x(y_1 z_2 - y_2 z_1) - y(x_1 z_2 - x_2 z_1) + z(x_1 y_2 - x_2 y_1) = 0
\]
