If `veca, vecb, vecc` are three non-zero non-null vectors are `vecr` is any vector in space then
`[(vecb, vecc, vecr)]veca+[(vecc, veca, vecr)]vecb+[(veca, vecb, vecr)]vecc` is equal to

Options

A. `2[(veca, vecb, vecc)]vecr`
B. `3[(veca, vecb, vecc)]vecr`
C. `[(veca, vecb, vecc)]`
D. None of these

Explanation

To solve the problem, we need to evaluate the expression:

\[ [(\vec{b}, \vec{c}, \vec{r})]\vec{a} + [(\vec{c}, \vec{a}, \vec{r})]\vec{b} + [(\vec{a}, \vec{b}, \vec{r})]\vec{c} \]

where \([\vec{x}, \vec{y}, \vec{z}]\) denotes the scalar triple product of vectors \(\vec{x}, \vec{y}, \vec{z}\).

### Step 1: Understand the Scalar Triple Product
The scalar triple product \([\vec{b}, \vec{c}, \vec{r}]\) can be expressed as:

\[ [\vec{b}, \vec{c}, \vec{r}] = \vec{b} \cdot (\vec{c} \times \vec{r}) \]

This gives us a scalar value which represents the volume of the parallelepiped formed by the vectors \(\vec{b}, \vec{c},\) and \(\vec{r}\).

### Step 2: Substitute the Scalar Triple Products
Now, we substitute the scalar triple products into the original expression:

\[ [\vec{b}, \vec{c}, \vec{r}]\vec{a} = \vec{a} \cdot (\vec{b} \times \vec{c}) \quad \text{(1)} \]
\[ [\vec{c}, \vec{a}, \vec{r}]\vec{b} = \vec{b} \cdot (\vec{c} \times \vec{a}) \quad \text{(2)} \]
\[ [\vec{a}, \vec{b}, \vec{r}]\vec{c} = \vec{c} \cdot (\vec{a} \times \vec{b}) \quad \text{(3)} \]

### Step 3: Combine the Results
Now, we combine the results from (1), (2), and (3):

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

### Step 4: Recognize the Identity
The expression we have is known as the scalar triple product identity, which states that:

\[ \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{c} \times \vec{a}) + \vec{c} \cdot (\vec{a} \times \vec{b}) = 0 \]

### Step 5: Conclusion
Thus, the final result of the expression is:

\[ \boxed{0} \]