What is equation of the plane if the foot of perpendicular from origin to this plane is (2,3,4) ?

Answer

To find the equation of the plane given that the foot of the perpendicular from the origin to the plane is the point \( (2, 3, 4) \), we can follow these steps:

### Step 1: Identify the foot of the perpendicular
The foot of the perpendicular from the origin \( O(0, 0, 0) \) to the plane is given as the point \( P(2, 3, 4) \).

### Step 2: Determine the direction vector
The direction vector \( \vec{n} \) (normal vector to the plane) can be represented by the vector from the origin to the point \( P \):
\[ \vec{OP} = P - O = (2, 3, 4) - (0, 0, 0) = 2\hat{i} + 3\hat{j} + 4\hat{k} \]
Thus, the normal vector \( \vec{n} = 2\hat{i} + 3\hat{j} + 4\hat{k} \).

### Step 3: Write the equation of the plane
The general equation of a plane can be expressed as:
\[ ax + by + cz = d \]
where \( (a, b, c) \) are the components of the normal vector \( \vec{n} \) and \( d \) is the dot product of the normal vector with the position vector of the foot of the perpendicular.

### Step 4: Calculate \( d \)
Using the coordinates of point \( P(2, 3, 4) \):
\[ d = \vec{n} \cdot \vec{OP} = (2, 3, 4) \cdot (2, 3, 4) = 2 \cdot 2 + 3 \cdot 3 + 4 \cdot 4 = 4 + 9 + 16 = 29 \]

### Step 5: Substitute into the plane equation
Now substituting \( a = 2 \), \( b = 3 \), \( c = 4 \), and \( d = 29 \) into the plane equation:
\[ 2x + 3y + 4z = 29 \]

### Final Answer
The equation of the plane is:
\[ 2x + 3y + 4z = 29 \]