The equation of the line throgh the point `veca` parallel to the plane `vecr.vecn`=q and perpendicular to the line `vecr=vecb+tvecc` is (A) `vecr=veca+lamda (vecnxxvecc)` (B) `(vecr-veca)xx(vecnxxvecc)=0` (C) `vecr=vecb+lamda(vecnxxvecc)` (D) none of these

Options

A. `vecr=veca+lamda(vecnxxvecc)`
B. `(vecr-veca)xx(vecnxxvecc)=0`
C. `vecr=vecb+lamda(vecnxxvecc)`
D. none of these

Explanation

To solve the problem, we need to find the equation of a line that satisfies two conditions: it passes through a given point \( \vec{a} \), is parallel to a specified plane, and is perpendicular to a given line. Let's break down the solution step by step.

### Step 1: Understand the conditions
The line we are looking for must:
1. Pass through the point \( \vec{a} \).
2. Be parallel to the plane defined by the equation \( \vec{r} \cdot \vec{n} = q \). This means it must be perpendicular to the normal vector \( \vec{n} \) of the plane.
3. Be perpendicular to the line defined by \( \vec{r} = \vec{b} + t \vec{c} \). This means it must be perpendicular to the direction vector \( \vec{c} \) of this line.

### Step 2: Determine the direction vector of the required line
Since the required line must be perpendicular to both \( \vec{n} \) and \( \vec{c} \), we can find its direction vector using the cross product:
\[ \text{Direction vector} = \vec{n} \times \vec{c} \]

### Step 3: Write the equation of the line
The general equation of a line in vector form that passes through a point \( \vec{a} \) and has a direction vector \( \vec{d} \) is given by:
\[ \vec{r} = \vec{a} + \lambda \vec{d} \]
Substituting our direction vector from Step 2, we have:
\[ \vec{r} = \vec{a} + \lambda (\vec{n} \times \vec{c}) \]

### Step 4: Finalize the equation
Thus, the equation of the required line is:
\[ \vec{r} = \vec{a} + \lambda (\vec{n} \times \vec{c}) \]

### Step 5: Check the options
Now, we compare this equation with the given options:
- (A) \( \vec{r} = \vec{a} + \lambda (\vec{n} \times \vec{c}) \) - This matches our derived equation.
- (B) \( (\vec{r} - \vec{a}) \times (\vec{n} \times \vec{c}) = 0 \) - This represents a different condition.
- (C) \( \vec{r} = \vec{b} + \lambda (\vec{n} \times \vec{c}) \) - This does not match.
- (D) None of these - Not applicable since we found a match.

### Conclusion
The correct answer is:
**(A) \( \vec{r} = \vec{a} + \lambda (\vec{n} \times \vec{c}) \)**