Find the equation of the plane through the points `A(2,2,-1),B(3,4,2)a n dC(7,0,6.)`

Options

A. `5x+2y+3z=17`
B. `5x+2y-3z=17`
C. `5x-2y+3z=17`
D. none of these

Explanation

To find the equation of the plane passing through the points \( A(2,2,-1) \), \( B(3,4,2) \), and \( C(7,0,6) \), we can follow these steps:

### Step 1: Find the vectors \( \vec{AB} \) and \( \vec{BC} \)
The vectors can be calculated as follows:

\[ \vec{AB} = B - A = (3 - 2, 4 - 2, 2 - (-1)) = (1, 2, 3) \]

\[ \vec{BC} = C - B = (7 - 3, 0 - 4, 6 - 2) = (4, -4, 4) \]

### Step 2: Find the normal vector \( \vec{n} \) using the cross product \( \vec{AB} \times \vec{BC} \)
The cross product can be calculated using the determinant of a matrix formed by the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of \( \vec{AB} \) and \( \vec{BC} \):

\[ \vec{n} = \vec{AB} \times \vec{BC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 4 & -4 & 4 \end{vmatrix} \]

Calculating the determinant:

\[ \vec{n} = \hat{i} \begin{vmatrix} 2 & 3 \\ -4 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 3 \\ 4 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 4 & -4 \end{vmatrix} \]

Calculating each of these determinants:

1. \( \hat{i} (2 \cdot 4 - 3 \cdot (-4)) = \hat{i} (8 + 12) = 20\hat{i} \)
2. \( -\hat{j} (1 \cdot 4 - 3 \cdot 4) = -\hat{j} (4 - 12) = 8\hat{j} \)
3. \( \hat{k} (1 \cdot (-4) - 2 \cdot 4) = \hat{k} (-4 - 8) = -12\hat{k} \)

So,

\[ \vec{n} = 20\hat{i} + 8\hat{j} - 12\hat{k} \]

### Step 3: Write the equation of the plane
The general equation of a plane can be expressed as:

\[ n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0 \]

Where \( (x_0, y_0, z_0) \) is a point on the plane (we can use point \( A(2, 2, -1) \)) and \( (n_1, n_2, n_3) \) are the components of the normal vector \( \vec{n} \).

Substituting the values:

\[ 20(x - 2) + 8(y - 2) - 12(z + 1) = 0 \]

Expanding this:

\[ 20x - 40 + 8y - 16 - 12z - 12 = 0 \]

Combining like terms:

\[ 20x + 8y - 12z - 68 = 0 \]

### Final Equation
The equation of the plane is:

\[ 20x + 8y - 12z = 68 \]