If `veca,vecb and vecc` are three non coplanar vectors and `vecr` is any vector in space, then `(vecaxxvecb),(vecrxxvecc)+(vecb xxvecc)xx(vecrxxveca)+(veccxxveca)xx(vecrxxvecb)=`

Options

A. (a)`2[veca vecb vec c] vec r`
B. (b)`3[veca vecb vec c]vec r`
C. (c)`[veca vecb vec c] vecr`
D. (d)None of these

Explanation

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

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

where \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are non-coplanar vectors, and \(\vec{r}\) is any vector in space.

### Step 1: Use the Vector Triple Product Identity

We will use the vector triple product identity, which states that:

\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]

This identity will help us simplify each term in our expression.

### Step 2: Simplify Each Term

1. **First Term**:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{r} \times \vec{c}) = \vec{c} \cdot (\vec{r} \times \vec{a} \times \vec{b}) = (\vec{c} \cdot \vec{r}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{r} \]

2. **Second Term**:
\[ (\vec{b} \times \vec{c}) \cdot (\vec{r} \times \vec{a}) = \vec{a} \cdot (\vec{r} \times \vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{r}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{r} \]

3. **Third Term**:
\[ (\vec{c} \times \vec{a}) \cdot (\vec{r} \times \vec{b}) = \vec{b} \cdot (\vec{r} \times \vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{r}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{r} \]

### Step 3: Combine All Terms

Now, we combine all three terms:

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

### Step 4: Group the Terms

Group the terms involving \(\vec{r}\):

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

### Step 5: Factor Out \(\vec{r}\)

This can be rewritten as:

\[ \vec{r} \cdot (\vec{a} \times \vec{b} \times \vec{c}) + \text{(terms with vectors)} \]

### Conclusion

The final result simplifies to:

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

Thus, we conclude that the expression evaluates to:

\[ \vec{a} \cdot \vec{b} \cdot \vec{c} \cdot \vec{r} \]