Four vectors `veca, vecb, vecc` and `vecx` satisfy the relation `(veca.vecx)vecb=vecc+vecx` where `vecb.veca ne 1`. The value of `vecx` in terms of `veca, vecb` and `vecc` is equal to

Options

A. `((veca.vecc)vecb-vecc(veca.vecb-1))/((veca.vecb-1))`
B. `vecc/(veca.vecb-1)`
C. `(2(veca.vecc)vecb+vecc)/(veca.vecb-1)`
D. `(2(veca.vecc)vecc+vecc)/((veca.vecb)-1)`

Explanation

To solve the problem, we need to find the vector \( \vec{x} \) in terms of the vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \) given the relation:

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

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ (\vec{a} \cdot \vec{x}) \vec{b} = \vec{c} + \vec{x} \]
Let's denote this as Equation (1).

2. **Take the dot product of both sides with \( \vec{a} \):**
\[ \vec{a} \cdot ((\vec{a} \cdot \vec{x}) \vec{b}) = \vec{a} \cdot \vec{c} + \vec{a} \cdot \vec{x} \]
The left-hand side can be simplified using the property of dot products:
\[ (\vec{a} \cdot \vec{x})(\vec{a} \cdot \vec{b}) = \vec{a} \cdot \vec{c} + \vec{a} \cdot \vec{x} \]

3. **Rearranging the equation:**
\[ (\vec{a} \cdot \vec{x})(\vec{a} \cdot \vec{b}) - \vec{a} \cdot \vec{x} = \vec{a} \cdot \vec{c} \]
Factor out \( \vec{a} \cdot \vec{x} \):
\[ \vec{a} \cdot \vec{x} \left( \vec{a} \cdot \vec{b} - 1 \right) = \vec{a} \cdot \vec{c} \]

4. **Solve for \( \vec{a} \cdot \vec{x} \):**
\[ \vec{a} \cdot \vec{x} = \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b} - 1} \]

5. **Substituting back into Equation (1):**
From Equation (1), we have:
\[ \vec{a} \cdot \vec{x} = \frac{\vec{c} + \vec{x}}{\vec{b}} \]
Setting the two expressions for \( \vec{a} \cdot \vec{x} \) equal:
\[ \frac{\vec{c} + \vec{x}}{\vec{b}} = \frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b} - 1} \]

6. **Cross-multiplying to eliminate the fraction:**
\[ (\vec{c} + \vec{x})(\vec{a} \cdot \vec{b} - 1) = \vec{b} \cdot \vec{a} \cdot \vec{c} \]

7. **Expanding and rearranging:**
\[ \vec{c}(\vec{a} \cdot \vec{b} - 1) + \vec{x}(\vec{a} \cdot \vec{b} - 1) = \vec{b} \cdot \vec{a} \cdot \vec{c} \]
Isolate \( \vec{x} \):
\[ \vec{x}(\vec{a} \cdot \vec{b} - 1) = \vec{b} \cdot \vec{a} \cdot \vec{c} - \vec{c}(\vec{a} \cdot \vec{b} - 1) \]

8. **Final expression for \( \vec{x} \):**
\[ \vec{x} = \frac{\vec{b} \cdot \vec{a} \cdot \vec{c} - \vec{c}(\vec{a} \cdot \vec{b} - 1)}{\vec{a} \cdot \vec{b} - 1} \]