If `veca , vecb , vec c ` are any three coplanar unit vectors , then :

Options

A. `veca. (vecb xx vec c ) = 1`
B. `veca . (vec b xx vec c ) = 3`
C. `(veca xx vec b ).vec c = 0`
D. `(vec c xx vec a ).vec b = 1`

Answer

To solve the problem, we need to show that for any three coplanar unit vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\).

### Step-by-step Solution:

1. **Understanding Coplanar Vectors**:
- Three vectors are said to be coplanar if they lie in the same plane. This means that any vector can be expressed as a linear combination of the other two.

2. **Definition of Scalar Triple Product**:
- The scalar triple product of three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is given by the expression \(\vec{a} \cdot (\vec{b} \times \vec{c})\).
- This scalar triple product can also be interpreted geometrically as the volume of the parallelepiped formed by the three vectors.

3. **Volume of Parallelepiped**:
- If the vectors are coplanar, the volume of the parallelepiped they form is zero because they do not span a three-dimensional space. Thus, the scalar triple product must also be zero.

4. **Calculating \(\vec{b} \times \vec{c}\)**:
- The vector \(\vec{b} \times \vec{c}\) is perpendicular to the plane formed by \(\vec{b}\) and \(\vec{c}\). Therefore, it will be orthogonal to any vector lying in that plane, including \(\vec{a}\).

5. **Dot Product with \(\vec{a}\)**:
- The dot product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) involves the cosine of the angle between \(\vec{a}\) and \(\vec{b} \times \vec{c}\). Since \(\vec{b} \times \vec{c}\) is perpendicular to the plane containing \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), the angle between \(\vec{a}\) and \(\vec{b} \times \vec{c}\) is \(90^\circ\).

6. **Conclusion**:
- The cosine of \(90^\circ\) is \(0\), thus \(\vec{a} \cdot (\vec{b} \times \vec{c}) = |\vec{a}| |\vec{b} \times \vec{c}| \cos(90^\circ) = 0\). Therefore, we conclude that:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]

### Final Result:
For any three coplanar unit vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), the scalar triple product is:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]