Let `veca,vecb` and `vecc` be three vectors. Then scalar triple product `[veca vecb vecc]` is equal to

Options

A. `[(vecb, veca, vecc)]`
B. `[veca vecc vecb]`
C. `[vecc vecb veca]`
D. `[vecb vecc veca]`

Explanation

To solve the problem regarding the scalar triple product of three vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \), we need to understand the properties of the scalar triple product.

### Step-by-Step Solution:

1. **Definition of Scalar Triple Product**:
The scalar triple product of three vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \) is defined as:
\[ [\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) \]
This represents the volume of the parallelepiped formed by the three vectors.

**Hint**: Remember that the scalar triple product can be computed using the dot product and the cross product.

2. **Cyclic Property**:
The scalar triple product has a cyclic property, which means that the order of the vectors can be changed in a cyclic manner without affecting the result. Thus:
\[ [\vec{a}, \vec{b}, \vec{c}] = [\vec{b}, \vec{c}, \vec{a}] = [\vec{c}, \vec{a}, \vec{b}] \]
This property is crucial when rearranging the vectors.

**Hint**: If you rearrange the vectors in a cyclic manner, the value of the scalar triple product remains unchanged.

3. **Anticommutative Property**:
If we swap any two vectors in the scalar triple product, the sign of the product changes. For example:
\[ [\vec{a}, \vec{b}, \vec{c}] = -[\vec{b}, \vec{a}, \vec{c}] \]
This property is important to remember when dealing with permutations of the vectors.

**Hint**: Swapping two vectors in the scalar triple product will change the sign of the result.

4. **Conclusion**:
Therefore, the scalar triple product \( [\vec{a}, \vec{b}, \vec{c}] \) is equal to the scalar triple product of any cyclic permutation of these vectors, such as \( [\vec{b}, \vec{c}, \vec{a}] \) or \( [\vec{c}, \vec{a}, \vec{b}] \).

**Final Result**:
\[ [\vec{a}, \vec{b}, \vec{c}] = [\vec{b}, \vec{c}, \vec{a}] = [\vec{c}, \vec{a}, \vec{b}] \]