The reflection of the point `veca` in the plane `vecr.vecn=q` is (A) `veca+ (vecq-veca.vecn)/(|vecn|` (B) `veca+2((vecq-veca.vecn)/(|vecn|^2))vecn` (C) `veca+(2(vecq+veca.vecn))/(|vecn|)` (D) none of these

Options

A. `veca+((vecq-veca.vecn))/(|vecn|)`
B. `veca+2(((vecq-veca.vecn))/(|vecn|^(2)))vecn`
C. `veca+(2(vecq-veca.vecn))/(|vecn|)vecn`
D. none of these

Explanation

To find the reflection of the point represented by the vector \(\vec{a}\) in the plane defined by the equation \(\vec{r} \cdot \vec{n} = q\), we can follow these steps:

### Step 1: Understand the Plane Equation
The equation of the plane is given by \(\vec{r} \cdot \vec{n} = q\), where \(\vec{n}\) is the normal vector to the plane and \(q\) is a scalar.

**Hint:** The normal vector \(\vec{n}\) is perpendicular to the plane.

### Step 2: Find the Projection of \(\vec{a}\) onto the Normal Vector
To find the projection of \(\vec{a}\) onto the normal vector \(\vec{n}\), we can use the formula:
\[ \text{Projection of } \vec{a} \text{ onto } \vec{n} = \frac{\vec{a} \cdot \vec{n}}{|\vec{n}|^2} \vec{n} \]

**Hint:** The dot product \(\vec{a} \cdot \vec{n}\) gives a scalar that helps in determining how far \(\vec{a}\) is from the plane along the direction of \(\vec{n}\).

### Step 3: Find the Point of Intersection
The point of intersection \(P\) of the line through \(\vec{a}\) in the direction of \(\vec{n}\) with the plane can be found by solving:
\[ \vec{a} + t\vec{n} \cdot \vec{n} = q \]
This gives us:
\[ t = \frac{q - \vec{a} \cdot \vec{n}}{|\vec{n}|^2} \]

**Hint:** Substitute \(t\) back into the line equation to find the coordinates of the intersection point.

### Step 4: Calculate the Reflection Point
The reflection point \(\vec{b}\) can be calculated using the formula:
\[ \vec{b} = \vec{a} + 2\left(\frac{q - \vec{a} \cdot \vec{n}}{|\vec{n}|^2}\right) \vec{n} \]

**Hint:** This formula accounts for moving from \(\vec{a}\) to the plane and then the same distance back, resulting in the reflection.

### Step 5: Identify the Correct Option
Now, we can compare our derived formula with the given options:
- The derived formula matches with option (B):
\[ \vec{b} = \vec{a} + 2\left(\frac{q - \vec{a} \cdot \vec{n}}{|\vec{n}|^2}\right) \vec{n} \]

Thus, the correct answer is **(B)**.

### Summary of Steps:
1. Understand the plane equation.
2. Find the projection of \(\vec{a}\) onto \(\vec{n}\).
3. Find the point of intersection with the plane.
4. Calculate the reflection point using the derived formula.
5. Identify the correct option based on the derived formula.