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. Reflection of `A(veca)` in the plane `pi` has the position vector :

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 find the reflection of point A (with position vector \(\vec{a}\)) in the plane \(\pi: \vec{r} \cdot \vec{n} = d\), we can follow these steps:

### Step 1: Understand the Plane Equation
The equation of the plane is given as \(\vec{r} \cdot \vec{n} = d\), where \(\vec{n}\) is the normal vector to the plane. This means that any point \(\vec{r}\) on the plane satisfies this equation.

### Step 2: Find the Projection of \(\vec{a}\) onto the Normal
To find the reflection, we first need to calculate the projection of the vector \(\vec{a}\) onto the normal vector \(\vec{n}\). The projection of \(\vec{a}\) onto \(\vec{n}\) is given by the formula:
\[ \text{Projection of } \vec{a} \text{ onto } \vec{n} = \frac{\vec{a} \cdot \vec{n}}{\|\vec{n}\|^2} \vec{n} \]

### Step 3: Calculate the Distance from Point A to the Plane
Next, we calculate the distance from point A to the plane. This distance can be found using the formula:
\[ \text{Distance} = \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|} \]

### Step 4: Determine the Point of Intersection
The point of intersection (let's call it \(\vec{p}\)) of the line through A perpendicular to the plane can be found by moving along the normal vector from point A:
\[ \vec{p} = \vec{a} - \text{Distance} \cdot \frac{\vec{n}}{\|\vec{n}\|} \]

### Step 5: Find the Reflection Point Q
The reflection point Q can be found by moving the same distance from the point of intersection \(\vec{p}\) to the other side of the plane:
\[ \vec{q} = \vec{p} + \text{Distance} \cdot \frac{\vec{n}}{\|\vec{n}\|} \]

### Step 6: Substitute and Simplify
Substituting the expression for \(\vec{p}\) into the equation for \(\vec{q}\):
\[ \vec{q} = \left( \vec{a} - \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|} \cdot \frac{\vec{n}}{\|\vec{n}\|} \right) + \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|} \cdot \frac{\vec{n}}{\|\vec{n}\|} \]

### Final Expression
After simplifying, we can express the reflection point \(\vec{q}\) in terms of \(\vec{a}\) and \(\vec{n}\):
\[ \vec{q} = \vec{a} - 2 \cdot \frac{\vec{a} \cdot \vec{n} - d}{\|\vec{n}\|^2} \cdot \vec{n} \]