If ` vec a` is perpendicular to ` vec b` and ` vec r` is non-zero vector such that `p vec r+( vec rdot vec a) vec b= vec c ,` then ` vec r=` ` vec c/p-(( vec adot vec c) vec b)/(p^2)` (b) ` vec a/p-(( vec cdot vec b) vec a)/(p^2)` ` vec a/p-(( vec adot vec b) vec c)/(p^2)` (d) ` vec c/(p^2)-(( vec adot vec c) vec b)/p`

Options

A. `[rac]=0`
B. `p^(2)r=pa-(c*a)b`
C. `p^(2)r=pb-(a*b)c`
D. `p^(2)r=pc-(b*c)a`

Answer

To solve the problem, we start with the given equation:

\[ p \vec{r} + (\vec{r} \cdot \vec{a}) \vec{b} = \vec{c} \]

### Step 1: Understand the Perpendicularity Condition
Since \(\vec{a}\) is perpendicular to \(\vec{b}\), we know that:

\[ \vec{a} \cdot \vec{b} = 0 \]

### Step 2: Take the Dot Product with \(\vec{a}\)
We take the dot product of both sides of the equation with \(\vec{a}\):

\[ \vec{a} \cdot (p \vec{r} + (\vec{r} \cdot \vec{a}) \vec{b}) = \vec{a} \cdot \vec{c} \]

### Step 3: Simplify the Left Side
Using the distributive property of the dot product, we have:

\[ p (\vec{a} \cdot \vec{r}) + (\vec{r} \cdot \vec{a})(\vec{a} \cdot \vec{b}) = \vec{a} \cdot \vec{c} \]

Since \(\vec{a} \cdot \vec{b} = 0\), the second term becomes zero:

\[ p (\vec{a} \cdot \vec{r}) = \vec{a} \cdot \vec{c} \]

### Step 4: Solve for \(\vec{r} \cdot \vec{a}\)
From the equation above, we can express \(\vec{r} \cdot \vec{a}\):

\[ \vec{r} \cdot \vec{a} = \frac{\vec{a} \cdot \vec{c}}{p} \]

### Step 5: Substitute Back into the Original Equation
Now substitute \(\vec{r} \cdot \vec{a}\) back into the original equation:

\[ p \vec{r} + \left(\frac{\vec{a} \cdot \vec{c}}{p}\right) \vec{b} = \vec{c} \]

### Step 6: Isolate \(\vec{r}\)
Rearranging gives us:

\[ p \vec{r} = \vec{c} - \left(\frac{\vec{a} \cdot \vec{c}}{p}\right) \vec{b} \]

### Step 7: Solve for \(\vec{r}\)
Now divide both sides by \(p\):

\[ \vec{r} = \frac{\vec{c}}{p} - \frac{\vec{a} \cdot \vec{c}}{p^2} \vec{b} \]

### Final Result
Thus, the expression for \(\vec{r}\) is:

\[ \vec{r} = \frac{\vec{c}}{p} - \frac{(\vec{a} \cdot \vec{c})}{p^2} \vec{b} \]

### Conclusion
The correct option is:

**(a)** \( \vec{r} = \frac{\vec{c}}{p} - \frac{(\vec{a} \cdot \vec{c}) \vec{b}}{p^2} \)