Let `vec r xx veca = vec b xx veca` and `vecc vecr=0`, where `veca.vecc ne 0`, then `veca.vecc(vecr xx vecb)+(vecb.vecc)(veca xx vecr)` is equal to __________.

Answer

To solve the problem, we need to analyze the given vector equations and manipulate them using vector identities.

### Step-by-Step Solution:

1. **Given Equations**: We start with the equations:
\[ \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \]
and
\[ \vec{c} \cdot \vec{r} = 0 \]
where \(\vec{a} \cdot \vec{c} \neq 0\).

2. **Cross Product Manipulation**: From the first equation, we can cross both sides with \(\vec{c}\):
\[ \vec{c} \times (\vec{r} \times \vec{a}) = \vec{c} \times (\vec{b} \times \vec{a}) \]

3. **Using Vector Triple Product Identity**: We apply the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
Thus, we have:
\[ \vec{c} \times (\vec{r} \times \vec{a}) = (\vec{c} \cdot \vec{a}) \vec{r} - (\vec{c} \cdot \vec{r}) \vec{a} \]
and
\[ \vec{c} \times (\vec{b} \times \vec{a}) = (\vec{c} \cdot \vec{a}) \vec{b} - (\vec{c} \cdot \vec{b}) \vec{a} \]

4. **Setting the Equations Equal**: Since both expressions are equal, we can set them equal:
\[ (\vec{c} \cdot \vec{a}) \vec{r} - 0 \cdot \vec{a} = (\vec{c} \cdot \vec{a}) \vec{b} - (\vec{c} \cdot \vec{b}) \vec{a} \]
Simplifying this gives:
\[ (\vec{c} \cdot \vec{a}) \vec{r} = (\vec{c} \cdot \vec{a}) \vec{b} - (\vec{c} \cdot \vec{b}) \vec{a} \]

5. **Rearranging the Terms**: Rearranging this equation leads to:
\[ \vec{r} = \vec{b} - \frac{\vec{c} \cdot \vec{b}}{\vec{c} \cdot \vec{a}} \vec{a} \]

6. **Finding the Required Expression**: Now we need to find:
\[ \vec{a} \cdot \vec{c} (\vec{r} \times \vec{b}) + (\vec{b} \cdot \vec{c})(\vec{a} \times \vec{r}) \]

7. **Substituting \(\vec{r}\)**: Substitute \(\vec{r}\) into the expression:
\[ \vec{a} \cdot \vec{c} \left(\left(\vec{b} - \frac{\vec{c} \cdot \vec{b}}{\vec{c} \cdot \vec{a}} \vec{a}\right) \times \vec{b}\right) + (\vec{b} \cdot \vec{c})\left(\vec{a} \times \left(\vec{b} - \frac{\vec{c} \cdot \vec{b}}{\vec{c} \cdot \vec{a}} \vec{a}\right)\right) \]

8. **Simplifying the Expression**: After simplification, we can see that the terms involving \(\vec{a} \times \vec{b}\) and \(\vec{b} \times \vec{a}\) will cancel out, leading to:
\[ 0 \]

### Final Answer:
Thus, the expression \(\vec{a} \cdot \vec{c} (\vec{r} \times \vec{b}) + (\vec{b} \cdot \vec{c})(\vec{a} \times \vec{r})\) is equal to:
\[ \boxed{0} \]