What is the equation of the plane passing through the point `(1,-1,-1)` and perpendicular to each of the planes `x-2y-8z=0 and 2x+5y-z=0`

Options

A. `7x-3y+2z=14`
B. `2x+5y-3z=12`
C. `x-7y+3z=4`
D. `14x-5y+3z=16`

Answer

To find the equation of the plane that passes through the point \( (1, -1, -1) \) and is perpendicular to the given planes \( x - 2y - 8z = 0 \) and \( 2x + 5y - z = 0 \), we can follow these steps:

### Step 1: Find the Normal Vectors of the Given Planes
The normal vector of a plane given by the equation \( Ax + By + Cz = D \) is \( \vec{n} = (A, B, C) \).

For the first plane \( x - 2y - 8z = 0 \):
- The normal vector \( \vec{n_1} = (1, -2, -8) \).

For the second plane \( 2x + 5y - z = 0 \):
- The normal vector \( \vec{n_2} = (2, 5, -1) \).

### Step 2: Find the Direction Ratios of the Required Plane
The required plane is perpendicular to both given planes. Therefore, its normal vector \( \vec{n} \) can be found by taking the cross product of \( \vec{n_1} \) and \( \vec{n_2} \).

\[ \vec{n} = \vec{n_1} \times \vec{n_2} = (1, -2, -8) \times (2, 5, -1) \]

Calculating the cross product:

\[ \vec{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & -8 \\ 2 & 5 & -1 \end{vmatrix} \]

Calculating the determinant:

\[ \vec{n} = \hat{i}((-2)(-1) - (-8)(5)) - \hat{j}((1)(-1) - (-8)(2)) + \hat{k}((1)(5) - (-2)(2)) \]

\[ = \hat{i}(2 + 40) - \hat{j}(-1 + 16) + \hat{k}(5 + 4) \]

\[ = \hat{i}(42) - \hat{j}(15) + \hat{k}(9) \]

Thus, the normal vector \( \vec{n} = (42, -15, 9) \).

### Step 3: Write the Equation of the Plane
The general equation of a plane with normal vector \( (a, b, c) \) passing through point \( (x_0, y_0, z_0) \) is given by:

\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]

Substituting \( a = 42, b = -15, c = 9 \) and the point \( (1, -1, -1) \):

\[ 42(x - 1) - 15(y + 1) + 9(z + 1) = 0 \]

Expanding this:

\[ 42x - 42 - 15y - 15 + 9z + 9 = 0 \]

Combining like terms:

\[ 42x - 15y + 9z - 48 = 0 \]

### Final Equation of the Plane
Thus, the equation of the plane is:

\[ 42x - 15y + 9z = 48 \]