If `veca, vecb, vecc` are three non coplanar, non zero vectors then `(veca.veca)(vecbxxvecc)+(veca.vecb)(veccxxveca)+(veca.vecc)(vecaxxvecb)` is equal to

Options

A. `[(veca, vecb, vecc)]vecc`
B. `[(vecb, vecc, veca)]veca`
C. `[(vecc, veca, vecb)]vecb`
D. none of these

Explanation

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

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

where \(\vec{a}, \vec{b}, \vec{c}\) are three non-coplanar, non-zero vectors.

### Step 1: Rewrite the expression using properties of dot and cross products

We can use the properties of the dot product and the cross product. Recall that the scalar triple product \((\vec{a} \cdot (\vec{b} \times \vec{c}))\) can be represented as the volume of the parallelepiped formed by the vectors \(\vec{a}, \vec{b}, \vec{c}\).

### Step 2: Group the terms

We can group the terms in the expression:

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

### Step 3: Use the scalar triple product identity

Recall the identity for the scalar triple product:

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

This means that we can express each of the terms in terms of the scalar triple product.

### Step 4: Substitute using the scalar triple product

Using the scalar triple product, we can rewrite the expression:

1. The first term becomes \((\vec{a} \cdot \vec{a}) (\vec{b} \times \vec{c})\).
2. The second term becomes \((\vec{a} \cdot \vec{b}) (\vec{c} \times \vec{a})\).
3. The third term becomes \((\vec{a} \cdot \vec{c}) (\vec{a} \times \vec{b})\).

### Step 5: Recognize that two vectors are the same

Notice that in the third term, \(\vec{a} \times \vec{a} = 0\). Thus, the third term vanishes.

### Step 6: Combine the remaining terms

Now we have:

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

### Step 7: Use the cyclic property of scalar triple product

By the cyclic property of the scalar triple product, we can express the remaining terms as:

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

### Conclusion

Thus, the final value of the expression is:

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

where \(V\) is the volume of the parallelepiped formed by \(\vec{a}, \vec{b}, \vec{c}\).