If `veca, vecb, vec c` are unit vectors, then the value of `|veca-2vecb|^(2)+|vecb-2vec c|^(2)+|vec c-2 vec a|^(2)` does not exceed to : (a) 9 (b) 12 (c) 18 (d) 21

Options

A. 9
B. 12
C. 18
D. 21

Answer

To solve the problem, we need to evaluate the expression

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

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

### Step 1: Expand each term using the formula for the square of a vector's magnitude

Using the identity \( |\vec{x} - \vec{y}|^2 = |\vec{x}|^2 + |\vec{y}|^2 - 2\vec{x} \cdot \vec{y} \), we can expand each term:

1. For \( |\vec{a} - 2\vec{b}|^2 \):
\[ |\vec{a} - 2\vec{b}|^2 = |\vec{a}|^2 + |2\vec{b}|^2 - 2\vec{a} \cdot (2\vec{b}) = 1 + 4 - 4\vec{a} \cdot \vec{b} \]
(since \(|\vec{a}|^2 = 1\) and \(|\vec{b}|^2 = 1\))

2. For \( |\vec{b} - 2\vec{c}|^2 \):
\[ |\vec{b} - 2\vec{c}|^2 = |\vec{b}|^2 + |2\vec{c}|^2 - 2\vec{b} \cdot (2\vec{c}) = 1 + 4 - 4\vec{b} \cdot \vec{c} \]

3. For \( |\vec{c} - 2\vec{a}|^2 \):
\[ |\vec{c} - 2\vec{a}|^2 = |\vec{c}|^2 + |2\vec{a}|^2 - 2\vec{c} \cdot (2\vec{a}) = 1 + 4 - 4\vec{c} \cdot \vec{a} \]

### Step 2: Combine the expanded terms

Now, we can combine all the expanded terms:

\[ |\vec{a} - 2\vec{b}|^2 + |\vec{b} - 2\vec{c}|^2 + |\vec{c} - 2\vec{a}|^2 = (1 + 4 - 4\vec{a} \cdot \vec{b}) + (1 + 4 - 4\vec{b} \cdot \vec{c}) + (1 + 4 - 4\vec{c} \cdot \vec{a}) \]

This simplifies to:

\[ 3 + 12 - 4(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 15 - 4(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \]

### Step 3: Analyze the dot product

The dot products \(\vec{a} \cdot \vec{b}\), \(\vec{b} \cdot \vec{c}\), and \(\vec{c} \cdot \vec{a}\) can take values between -1 and 1.

To find the maximum value of the expression, we consider the minimum possible value of the sum of the dot products:

\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \geq -\frac{3}{2} \]

This leads to:

\[ 15 - 4\left(-\frac{3}{2}\right) = 15 + 6 = 21 \]

### Conclusion

Thus, the value of

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

does not exceed 21.

The correct answer is (d) 21.