If `r.n=q` is the equation of a plane normal to the vector n, the length of the perpendicular from the origin on the plane is

Options

A. q
B. `|n|`
C. `q|n|`
D. `(q)/(|n|)`

Answer

To find the length of the perpendicular from the origin to the plane given by the equation \( r \cdot n = q \), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Equation of the Plane**:
The equation \( r \cdot n = q \) represents a plane where \( r \) is the position vector of a point on the plane, \( n \) is a normal vector to the plane, and \( q \) is a scalar.

2. **Identify the Normal Vector**:
The normal vector \( n \) can be expressed in terms of its magnitude. We denote the magnitude of \( n \) as \( |n| \).

3. **Define the Unit Normal Vector**:
The unit normal vector \( \hat{n} \) is given by:
\[ \hat{n} = \frac{n}{|n|} \]

4. **Length of the Perpendicular**:
The length of the perpendicular from the origin to the plane can be represented as \( p \). The formula for the length of the perpendicular from a point to a plane is given by:
\[ p = \frac{|r_0 \cdot n - q|}{|n|} \]
where \( r_0 \) is the position vector of the point from which we are measuring the perpendicular distance. In this case, \( r_0 \) is the origin, which has coordinates (0, 0, 0), so \( r_0 \cdot n = 0 \).

5. **Substituting Values**:
Therefore, substituting into the formula, we have:
\[ p = \frac{|0 \cdot n - q|}{|n|} = \frac{| - q |}{|n|} = \frac{q}{|n|} \]

6. **Conclusion**:
Thus, the length of the perpendicular from the origin to the plane is:
\[ p = \frac{q}{|n|} \]