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

Options

A. `0`
B. `2`
C. `-2`
D. None of these

Answer

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

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

### Step 1: Recognize the Scalar Triple Product
The term \( a \cdot (b \times c) \) can be recognized as the scalar triple product, which is denoted as \( [a, b, c] \). Therefore, we can rewrite the first term:

\[ a \cdot (b \times c) = [a, b, c] \]

### Step 2: Rewrite the Expression
Now we can rewrite the entire expression using the scalar triple product notation:

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

### Step 3: Recognize the Second Scalar Triple Product
The term \( b \cdot (c \times a) \) is also a scalar triple product, which can be rewritten as:

\[ b \cdot (c \times a) = [b, c, a] \]

### Step 4: Rewrite the Second Term
Now, we can rewrite the second term in the expression:

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

### Step 5: Combine the Terms
Now we can combine the two terms:

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

### Step 6: Use Properties of Scalar Triple Product
Using the properties of scalar triple products, we know that:

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

Thus, we can substitute this into our expression:

\[ [a, b, c] + [b, c, a] = [a, b, c] - [a, b, c] = 0 \]

### Step 7: Final Result
Now substituting back into our expression, we have:

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

Therefore, the value of the given expression is:

\[ \boxed{0} \]