If `[veca vecb vecc]=1` then value of `(veca.vecbxxvecc)/(veccxxveca.vecb)+(vecb.veccxxveca)/(vecaxxvecb.vecc)+(vecc.vecaxxvecb)/(vecbxxvecc.veca)` is

Options

A. `3`
B. `1`
C. `-1`
D. None of these

Explanation

To solve the problem, we need to evaluate the expression given the condition that the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ] = 1\).

The expression we need to evaluate is:

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

### Step 1: Identify the Scalar Triple Products
Recall that the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) can be expressed in terms of dot and cross products:
- \([ \vec{a}, \vec{b}, \vec{c} ] = \vec{a} \cdot (\vec{b} \times \vec{c})\)
- \([ \vec{b}, \vec{c}, \vec{a} ] = \vec{b} \cdot (\vec{c} \times \vec{a})\)
- \([ \vec{c}, \vec{a}, \vec{b} ] = \vec{c} \cdot (\vec{a} \times \vec{b})\)

Given that \([ \vec{a}, \vec{b}, \vec{c} ] = 1\), we can say:
- \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 1\)
- \(\vec{b} \cdot (\vec{c} \times \vec{a}) = 1\)
- \(\vec{c} \cdot (\vec{a} \times \vec{b}) = 1\)

### Step 2: Substitute the Scalar Triple Products into the Expression
Now we can substitute these values into the expression:

\[ \frac{1}{\vec{c} \times \vec{a} \cdot \vec{b}} + \frac{1}{\vec{a} \times \vec{b} \cdot \vec{c}} + \frac{1}{\vec{b} \times \vec{c} \cdot \vec{a}} \]

### Step 3: Recognize the Denominators
Next, we notice that:
- \(\vec{c} \times \vec{a} \cdot \vec{b} = [ \vec{c}, \vec{a}, \vec{b} ] = 1\)
- \(\vec{a} \times \vec{b} \cdot \vec{c} = [ \vec{a}, \vec{b}, \vec{c} ] = 1\)
- \(\vec{b} \times \vec{c} \cdot \vec{a} = [ \vec{b}, \vec{c}, \vec{a} ] = 1\)

### Step 4: Substitute the Values into the Expression
Substituting these values into the expression gives us:

\[ \frac{1}{1} + \frac{1}{1} + \frac{1}{1} = 1 + 1 + 1 = 3 \]

### Final Answer
Thus, the value of the given expression is:

\[ \boxed{3} \]