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. `(|(vecb-veca)*vecn|)/(|vecn|)`
B. `|(vecb-veca)*vecn|`
C. `(|(vecb-veca)xx vecn|)/(|vecn|)`
D. `|(vecb-veca)xxvecn|`

Answer

To find the length of the segment \( PQ \) (the distance between the feet of the perpendiculars from points \( A \) and \( B \) to the plane \( \pi \)), we can follow these steps:

### Step 1: Understand the Plane Equation
The plane is given by the equation:
\[ \vec{r} \cdot \vec{n} = d \]
where \( \vec{n} \) is the normal vector to the plane and \( d \) is a constant.

### Step 2: Identify Points A and B
Let \( \vec{A} \) and \( \vec{B} \) be the position vectors of points \( A \) and \( B \) respectively. Both points lie on the same side of the plane.

### Step 3: Find the Foot of the Perpendiculars
The foot of the perpendicular from point \( A \) to the plane is denoted as point \( P \) and from point \( B \) to the plane as point \( Q \).

### Step 4: Calculate the Vector \( \vec{AB} \)
The vector from point \( A \) to point \( B \) is given by:
\[ \vec{AB} = \vec{B} - \vec{A} \]

### Step 5: Find the Projection of \( \vec{AB} \) onto the Normal Vector
To find the length of \( PQ \), we need to project \( \vec{AB} \) onto the direction of the normal vector \( \vec{n} \). The projection of \( \vec{AB} \) onto \( \vec{n} \) is given by:
\[ \text{Projection of } \vec{AB} \text{ on } \vec{n} = \frac{\vec{AB} \cdot \vec{n}}{|\vec{n}|^2} \vec{n} \]

### Step 6: Calculate the Length of the Segment \( PQ \)
The length \( PQ \) can be calculated as the magnitude of the projection of \( \vec{AB} \) onto \( \vec{n} \):
\[ PQ = \left| \frac{\vec{AB} \cdot \vec{n}}{|\vec{n}|} \right| \]

### Final Expression
Thus, the length of \( PQ \) is given by:
\[ PQ = \frac{|\vec{B} - \vec{A} \cdot \vec{n}|}{|\vec{n}|} \]

### Summary
The length of the segment \( PQ \) is determined by the projection of the vector \( \vec{AB} \) onto the normal vector \( \vec{n} \) of the plane.

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