STATEMENT-1: If `veca,vecb,vecc`, are unit vectors such that `veca+vecb+vecc=0` and `veca.vecb+vecb.vecc+vecc.veca=-(3)/(2)`.
STATEMENT-2 : `(vecx+vecy)^(2)=|vecx|^(2)+|vecy|^(2)+2(vecx.vecy)`.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-5
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-5
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the given statements and prove their validity step by step.

### Step 1: Analyze Statement 1
We are given three unit vectors \( \vec{a}, \vec{b}, \vec{c} \) such that:
\[ \vec{a} + \vec{b} + \vec{c} = 0 \]
This implies that:
\[ \vec{c} = -(\vec{a} + \vec{b}) \]

### Step 2: Calculate the Magnitudes
Since \( \vec{a}, \vec{b}, \vec{c} \) are unit vectors, we have:
\[ |\vec{a}| = |\vec{b}| = |\vec{c}| = 1 \]
Thus:
\[ |\vec{c}|^2 = 1 \implies \vec{c} \cdot \vec{c} = 1 \]

### Step 3: Expand the Equation
We can expand \( |\vec{c}|^2 \):
\[ \vec{c} \cdot \vec{c} = (-\vec{a} - \vec{b}) \cdot (-\vec{a} - \vec{b}) = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) \]
This gives:
\[ \vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 1 \]
Since \( \vec{a} \cdot \vec{a} = 1 \) and \( \vec{b} \cdot \vec{b} = 1 \), we have:
\[ 1 + 2\vec{a} \cdot \vec{b} + 1 = 1 \]
Simplifying this, we get:
\[ 2 + 2\vec{a} \cdot \vec{b} = 1 \implies 2\vec{a} \cdot \vec{b} = -1 \implies \vec{a} \cdot \vec{b} = -\frac{1}{2} \]

### Step 4: Calculate the Other Dot Products
Using symmetry and the fact that the vectors are unit vectors, we can derive:
\[ \vec{b} \cdot \vec{c} = \vec{b} \cdot (-\vec{a} - \vec{b}) = -\vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} = -(-\frac{1}{2}) - 1 = \frac{1}{2} - 1 = -\frac{1}{2} \]
Similarly:
\[ \vec{c} \cdot \vec{a} = \vec{c} \cdot (-\vec{a} - \vec{b}) = -\vec{c} \cdot \vec{a} - \vec{c} \cdot \vec{b} = -(-\frac{1}{2}) - (-\frac{1}{2}) = \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 5: Sum of Dot Products
Now we can sum these dot products:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -\frac{1}{2} - \frac{1}{2} + 1 = -\frac{3}{2} \]
This confirms that Statement 1 is true.

### Step 6: Analyze Statement 2
The second statement is a well-known vector identity:
\[ (\vec{x} + \vec{y})^2 = |\vec{x}|^2 + |\vec{y}|^2 + 2(\vec{x} \cdot \vec{y}) \]
This is always true for any vectors \( \vec{x} \) and \( \vec{y} \).

### Conclusion
Both statements are true, and Statement 2 provides a correct explanation for Statement 1.