Let `veca, vecb` and `vecc` are three unit vectors in a plane such that they are equally inclined to each other, then the value of `(veca xx vecb).(vecb xx vecc) + (vecb xx vecc). (vecc xx veca)+(vecc xx veca). (veca xx vecb)` can be

Options

A. `9/4`
B. `-9/4`
C. `3/4`
D. `-3/4`

Explanation

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

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

Given that \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors that are equally inclined to each other, the angle between any two vectors is \(120^\circ\).

### Step 1: Understanding the Cross Products
Since \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and equally inclined, we can use the properties of cross products. The magnitude of the cross product of two vectors is given by:

\[ |\vec{u} \times \vec{v}| = |\vec{u}| |\vec{v}| \sin(\theta) \]

where \(\theta\) is the angle between the vectors. For unit vectors, \(|\vec{u}| = |\vec{v}| = 1\).

### Step 2: Calculate Each Term
Let's calculate the first term \((\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c})\):

1. **Magnitude of \(\vec{a} \times \vec{b}\)**:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(120^\circ) = 1 \cdot 1 \cdot \sin(120^\circ) = \sin(120^\circ) = \frac{\sqrt{3}}{2} \]

2. **Magnitude of \(\vec{b} \times \vec{c}\)**:
\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin(120^\circ) = 1 \cdot 1 \cdot \sin(120^\circ) = \frac{\sqrt{3}}{2} \]

3. **Angle between \(\vec{a} \times \vec{b}\) and \(\vec{b} \times \vec{c}\)**:
Since \(\vec{a}, \vec{b}, \vec{c}\) are in the same plane, \(\vec{a} \times \vec{b}\) is perpendicular to the plane formed by \(\vec{a}\) and \(\vec{b}\), and similarly for \(\vec{b} \times \vec{c}\). Therefore, the angle between these two cross products is \(0^\circ\).

4. **Dot Product**:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) = |\vec{a} \times \vec{b}| |\vec{b} \times \vec{c}| \cos(0^\circ) = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \cdot 1 = \frac{3}{4} \]

### Step 3: Repeat for Other Terms
By symmetry, we can assert that:

\[ (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) = \frac{3}{4} \]
\[ (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = \frac{3}{4} \]

### Step 4: Sum the Terms
Now, we can sum all the terms:

\[ (\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = \frac{3}{4} + \frac{3}{4} + \frac{3}{4} = \frac{9}{4} \]

### Final Answer
Thus, the value of the expression is:

\[ \frac{9}{4} \]