If a, b and c are three non-coplanar vectors, then find the value of `(a*(btimesc))/(ctimes(a*b))+(b*(ctimesa))/(c*(atimesb))`.

Options

A. a) `0`
B. b) `2`
C. c) `-2`
D. d) None of these

Answer

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

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

where \(a\), \(b\), and \(c\) are three non-coplanar vectors.

### Step 1: Rewrite the Dot and Cross Products
We can express the dot and cross products in terms of the scalar triple product. The scalar triple product of vectors \(x\), \(y\), and \(z\) is given by \(x \cdot (y \times z)\) and can be denoted as \([x, y, z]\).

Thus, we can rewrite the terms in the expression:
- \(a \cdot (b \times c) = [a, b, c]\)
- \(b \cdot (c \times a) = [b, c, a]\)

### Step 2: Rewrite the Denominators
Now, we rewrite the denominators:
- \(c \times (a \cdot b)\) can be expressed as \([c, a, b]\)
- \(c \cdot (a \times b) = [c, a, b]\)

### Step 3: Substitute the Scalar Triple Products
Now we substitute these into our expression:

\[ \frac{[a, b, c]}{[c, a, b]} + \frac{[b, c, a]}{[c, a, b]} \]

### Step 4: Combine the Terms
Since both terms have the same denominator, we can combine them:

\[ \frac{[a, b, c] + [b, c, a]}{[c, a, b]} \]

### Step 5: Evaluate the Numerator
Now, we need to evaluate the numerator. Notice that:

\[ [a, b, c] + [b, c, a] = [a, b, c] + [b, c, a] = [a, b, c] + [c, a, b] = 2[a, b, c] \]

### Step 6: Final Expression
Thus, we can simplify the expression to:

\[ \frac{2[a, b, c]}{[c, a, b]} \]

### Step 7: Simplifying Further
Since the scalar triple product is invariant under cyclic permutations, we have:

\[ [c, a, b] = -[a, b, c] \]

Thus, we can rewrite the expression as:

\[ \frac{2[a, b, c]}{-[a, b, c]} = -2 \]

### Conclusion
Therefore, the final value of the expression is:

\[ \boxed{-2} \]