If `veca, vecb, vecc` are non-coplanar vectors, then `(veca.(vecb xx vecc))/(vecb.(vecc xx veca)) + (vecb.(vecc xx veca))/(vecc.(veca xx vecb)) +(vecc.(vecb xx veca))/(veca. (vecb xx vecc))` is equal to:

Options

A. 0
B. 4
C. 5
D. 6

Answer

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

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

### Step 1: Identify Scalar Triple Products
We recognize that the dot product of a vector with the cross product of two other vectors is known as the scalar triple product. We denote this as:

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

### Step 2: Rewrite Each Term
Using the property of scalar triple products, we can rewrite the terms in the expression:

1. \(\vec{b} \cdot (\vec{c} \times \vec{a}) = [\vec{b}, \vec{c}, \vec{a}] = -[\vec{a}, \vec{b}, \vec{c}]\)
2. \(\vec{c} \cdot (\vec{a} \times \vec{b}) = [\vec{c}, \vec{a}, \vec{b}] = -[\vec{b}, \vec{c}, \vec{a}]\)
3. \(\vec{a} \cdot (\vec{b} \times \vec{c}) = [\vec{a}, \vec{b}, \vec{c}]\)

### Step 3: Substitute Back into the Expression
Now we substitute these back into the original expression:

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

### Step 4: Simplify Each Fraction
This simplifies to:

\[ -1 + 1 + 1 \]

### Step 5: Calculate the Final Result
Now, we can calculate the final result:

\[ -1 + 1 + 1 = 1 \]

Thus, the final answer is:

\[ \boxed{1} \]