If `veca` is perpendicular to `vecb` and `vecr` is a non-zero vector such that `pvecr + (vecr.vecb)veca=vecc`, then `vecr` is:

Options

A. `vecc/p-((vecb.vecc)veca)/p^(2)`
B. `veca/p-((vecc.veca)vecb)/p^(2)`
C. `vecb/p-((veca.vecb)vecc)/p^(2)`
D. `vecc/p-((vecb.vecc)veca)/p^(2)`

Answer

To solve the problem, we need to find the vector \(\vec{r}\) given the conditions that \(\vec{a}\) is perpendicular to \(\vec{b}\) and the equation \(p\vec{r} + (\vec{r} \cdot \vec{b})\vec{a} = \vec{c}\).

### Step-by-Step Solution:

1. **Understand the given conditions**:
- Since \(\vec{a}\) is perpendicular to \(\vec{b}\), we have:
\[ \vec{a} \cdot \vec{b} = 0 \]

2. **Write the given equation**:
- The equation provided is:
\[ p\vec{r} + (\vec{r} \cdot \vec{b})\vec{a} = \vec{c} \]

3. **Rearranging the equation**:
- We can rearrange the equation to isolate \(p\vec{r}\):
\[ p\vec{r} = \vec{c} - (\vec{r} \cdot \vec{b})\vec{a} \]

4. **Express \(\vec{r}\)**:
- Dividing both sides by \(p\) gives:
\[ \vec{r} = \frac{\vec{c}}{p} - \frac{(\vec{r} \cdot \vec{b})\vec{a}}{p} \]

5. **Finding \(\vec{r} \cdot \vec{b}\)**:
- To find \(\vec{r} \cdot \vec{b}\), we take the dot product of both sides of the rearranged equation with \(\vec{b}\):
\[ \vec{b} \cdot (p\vec{r}) = \vec{b} \cdot \vec{c} - \vec{b} \cdot ((\vec{r} \cdot \vec{b})\vec{a}) \]
- This simplifies to:
\[ p(\vec{r} \cdot \vec{b}) = \vec{b} \cdot \vec{c} - (\vec{r} \cdot \vec{b})(\vec{b} \cdot \vec{a}) \]
- Since \(\vec{a} \cdot \vec{b} = 0\), the second term vanishes:
\[ p(\vec{r} \cdot \vec{b}) = \vec{b} \cdot \vec{c} \]

6. **Solving for \(\vec{r} \cdot \vec{b}\)**:
- Thus, we have:
\[ \vec{r} \cdot \vec{b} = \frac{\vec{b} \cdot \vec{c}}{p} \]

7. **Substituting back into the expression for \(\vec{r}\)**:
- Now substitute \(\vec{r} \cdot \vec{b}\) back into the expression for \(\vec{r}\):
\[ \vec{r} = \frac{\vec{c}}{p} - \frac{\left(\frac{\vec{b} \cdot \vec{c}}{p}\right)\vec{a}}{p} \]
- This simplifies to:
\[ \vec{r} = \frac{\vec{c}}{p} - \frac{\vec{b} \cdot \vec{c}}{p^2}\vec{a} \]

8. **Final expression for \(\vec{r}\)**:
- Therefore, the final expression for \(\vec{r}\) is:
\[ \vec{r} = \frac{\vec{c}}{p} - \frac{\vec{b} \cdot \vec{c}}{p^2}\vec{a} \]