Consider a plane `pi:vecr*vecn=d` (where `vecn` is not a unti vector). There are two points `A(veca)` and `B(vecb)` lying on the same side of the plane.
Q. If foot of perpendicular from A and B to the plane `pi` are P and Q respectively, then length of PQ be :

Options

A. `veca+(2)/((vecn)^(2))(d-veca*vecn)vecn`
B. `veca-(1)/((vecn)^(2))(d-veca*vecn)vecn`
C. `veca+(2)/((vecn)^(2))(d+veca*vecn)vecn`
D. `veca+(2)/((vecn)^(2))vecn`

Answer

To solve the problem, we need to find the length of the segment \( PQ \) where \( P \) and \( Q \) are the feet of the perpendiculars dropped from points \( A \) and \( B \) to the plane \( \pi \) defined by the equation \( \vec{r} \cdot \vec{n} = d \).

### Step-by-step Solution:

1. **Understanding the Plane and Points**:
The plane is defined by the equation \( \vec{r} \cdot \vec{n} = d \), where \( \vec{n} \) is the normal vector to the plane. Points \( A \) and \( B \) are given as vectors \( \vec{a} \) and \( \vec{b} \), respectively, and both lie on the same side of the plane.

2. **Finding the Foot of Perpendiculars**:
The foot of the perpendicular from point \( A \) to the plane is denoted as \( P \), and from point \( B \) to the plane is denoted as \( Q \). The position vectors of these feet can be found using the formula for the foot of the perpendicular from a point to a plane.

The foot of the perpendicular from point \( A \) to the plane is given by:
\[ \vec{p} = \vec{a} - \frac{(\vec{a} \cdot \vec{n} - d)}{|\vec{n}|^2} \vec{n} \]
Similarly, for point \( B \):
\[ \vec{q} = \vec{b} - \frac{(\vec{b} \cdot \vec{n} - d)}{|\vec{n}|^2} \vec{n} \]

3. **Finding the Length of Segment \( PQ \)**:
The length of segment \( PQ \) can be calculated using the distance formula:
\[ PQ = |\vec{p} - \vec{q}| \]
Substituting the expressions for \( \vec{p} \) and \( \vec{q} \):
\[ PQ = \left| \left( \vec{a} - \frac{(\vec{a} \cdot \vec{n} - d)}{|\vec{n}|^2} \vec{n} \right) - \left( \vec{b} - \frac{(\vec{b} \cdot \vec{n} - d)}{|\vec{n}|^2} \vec{n} \right) \right| \]
Simplifying this expression:
\[ PQ = \left| \vec{a} - \vec{b} + \frac{(\vec{b} \cdot \vec{n} - d) - (\vec{a} \cdot \vec{n} - d)}{|\vec{n}|^2} \vec{n} \right| \]
This can be rewritten as:
\[ PQ = |\vec{b} - \vec{a} + \frac{(\vec{b} - \vec{a}) \cdot \vec{n}}{|\vec{n}|^2} \vec{n}| \]

4. **Final Expression**:
To find the length \( PQ \), we can express it in terms of the cross product:
\[ PQ = \frac{|\vec{b} - \vec{a} \times \vec{n}|}{|\vec{n}|} \]
This gives us the required length of segment \( PQ \).

### Final Answer:
The length of \( PQ \) is given by:
\[ PQ = \frac{|\vec{b} - \vec{a} \times \vec{n}|}{|\vec{n}|} \]