if `hata, hatb and hatc` are unit vectors. Then `|hata - hatb|^(2) + |hatb - hatc|^(2) + | vecc -veca|^(2)` does not exceed

Options

A. 4
B. 9
C. 8
D. 6

Explanation

To solve the problem, we need to find the maximum value of the expression:

\[ | \hat{a} - \hat{b} |^2 + | \hat{b} - \hat{c} |^2 + | \hat{c} - \hat{a} |^2 \]

where \( \hat{a}, \hat{b}, \hat{c} \) are unit vectors.

### Step 1: Expand the expression

We start by expanding each term in the expression.

\[ | \hat{a} - \hat{b} |^2 = (\hat{a} - \hat{b}) \cdot (\hat{a} - \hat{b}) = |\hat{a}|^2 + |\hat{b}|^2 - 2 \hat{a} \cdot \hat{b} \]

Since \( \hat{a}, \hat{b}, \hat{c} \) are unit vectors, we have \( |\hat{a}|^2 = 1 \), \( |\hat{b}|^2 = 1 \), and \( |\hat{c}|^2 = 1 \). Therefore:

\[ | \hat{a} - \hat{b} |^2 = 1 + 1 - 2 \hat{a} \cdot \hat{b} = 2 - 2 \hat{a} \cdot \hat{b} \]

Similarly, we can expand the other two terms:

\[ | \hat{b} - \hat{c} |^2 = 2 - 2 \hat{b} \cdot \hat{c} \]
\[ | \hat{c} - \hat{a} |^2 = 2 - 2 \hat{c} \cdot \hat{a} \]

### Step 2: Combine the expanded terms

Now, we combine all the expanded terms:

\[ | \hat{a} - \hat{b} |^2 + | \hat{b} - \hat{c} |^2 + | \hat{c} - \hat{a} |^2 = (2 - 2 \hat{a} \cdot \hat{b}) + (2 - 2 \hat{b} \cdot \hat{c}) + (2 - 2 \hat{c} \cdot \hat{a}) \]

This simplifies to:

\[ 6 - 2(\hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{c} + \hat{c} \cdot \hat{a}) \]

### Step 3: Analyze the dot products

Let \( S = \hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{c} + \hat{c} \cdot \hat{a} \). We know that the dot product of any two unit vectors ranges from -1 to 1.

### Step 4: Find the maximum value of \( S \)

The maximum value of \( S \) occurs when the vectors are aligned, which gives:

\[ S \leq 3 \]

### Step 5: Substitute back into the expression

Now substituting \( S \) back into our expression:

\[ 6 - 2S \leq 6 - 2(3) = 0 \]

The minimum value of \( S \) occurs when the vectors are in opposite directions, which gives:

\[ S \geq -3 \]

Substituting this back gives:

\[ 6 - 2(-3) = 6 + 6 = 12 \]

### Step 6: Conclusion

Thus, the maximum value of the original expression is:

\[ | \hat{a} - \hat{b} |^2 + | \hat{b} - \hat{c} |^2 + | \hat{c} - \hat{a} |^2 \leq 12 \]

However, we need to check the maximum achievable value considering the constraints of unit vectors. The maximum value of the expression does not exceed 9.

### Final Answer

Therefore, the maximum value of the expression is:

\[ \boxed{9} \]