The foot of the perpendicular drawn from the origin to the plane is the point (2, 5, 7). Find the equation of the plane.

Answer

To find the equation of the plane given that the foot of the perpendicular drawn from the origin to the plane is the point (2, 5, 7), we can follow these steps:

### Step 1: Identify the point on the plane
The foot of the perpendicular from the origin (0, 0, 0) to the plane is given as the point \( P(2, 5, 7) \). This point lies on the plane.

### Step 2: Determine the normal vector
The normal vector \( \mathbf{n} \) of the plane can be found by taking the vector from the origin to the point \( P \). This vector is given by:
\[ \mathbf{n} = \mathbf{OP} = (2 - 0, 5 - 0, 7 - 0) = (2, 5, 7) \]

### Step 3: Write the equation of the plane
The general equation of a plane in 3D space 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, and \( (n_1, n_2, n_3) \) is the normal vector.

Substituting \( (x_0, y_0, z_0) = (2, 5, 7) \) and \( (n_1, n_2, n_3) = (2, 5, 7) \):
\[ 2(x - 2) + 5(y - 5) + 7(z - 7) = 0 \]

### Step 4: Expand the equation
Expanding this equation:
\[ 2x - 4 + 5y - 25 + 7z - 49 = 0 \]
Combining like terms:
\[ 2x + 5y + 7z - 78 = 0 \]

### Step 5: Rearranging to standard form
Rearranging gives us the equation of the plane:
\[ 2x + 5y + 7z = 78 \]

### Final Answer
The equation of the plane is:
\[ 2x + 5y + 7z = 78 \]

---