If `veca, vecb, vec c ` are three non coplanar vectors , then the value of `(vec a.(vec b xx vec c) )/((vec c xx vec a).vec b) + ( vecb.(vec a xx vec c ))/(vec c.(vec a xx vec b))`is

Options

A. 0
B. 2
C. 1
D. none of these

Answer

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

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

### Step 1: Recognize the Scalar Triple Product
The term \(\vec{a} \cdot (\vec{b} \times \vec{c})\) is known as the scalar triple product, which can be denoted as \([\vec{a}, \vec{b}, \vec{c}]\). This scalar triple product represents the volume of the parallelepiped formed by the vectors \(\vec{a}, \vec{b}, \vec{c}\).

### Step 2: Apply the Cyclic Property
The scalar triple product has a cyclic property:
\[ [\vec{a}, \vec{b}, \vec{c}] = [\vec{b}, \vec{c}, \vec{a}] = [\vec{c}, \vec{a}, \vec{b}] \]
Thus, we can express the terms in the numerator using this property.

### Step 3: Rewrite the Denominators
The denominator \((\vec{c} \times \vec{a}) \cdot \vec{b}\) can also be expressed as \([\vec{b}, \vec{c}, \vec{a}]\) and \(\vec{c} \cdot (\vec{a} \times \vec{b})\) as \([\vec{a}, \vec{b}, \vec{c}]\).

### Step 4: Substitute into the Expression
Substituting these into our expression gives:
\[ \frac{[\vec{a}, \vec{b}, \vec{c}]}{[\vec{b}, \vec{c}, \vec{a}]} + \frac{[\vec{b}, \vec{a}, \vec{c}]}{[\vec{a}, \vec{b}, \vec{c}]} \]

### Step 5: Simplify the Expression
Using the cyclic property:
1. The first term becomes \(\frac{[\vec{a}, \vec{b}, \vec{c}]}{[\vec{b}, \vec{c}, \vec{a}]} = 1\) because \([\vec{b}, \vec{c}, \vec{a}] = [\vec{a}, \vec{b}, \vec{c}]\).
2. The second term simplifies to \(\frac{[\vec{b}, \vec{a}, \vec{c}]}{[\vec{a}, \vec{b}, \vec{c}]} = -1\) because switching two vectors in the scalar triple product changes its sign.

### Step 6: Combine the Results
Thus, we have:
\[ 1 + (-1) = 0 \]

### Final Result
The value of the given expression is:
\[ \boxed{0} \]