If `veca and vecb` are non - zero vectors such that `|veca + vecb| = |veca - 2vecb|` then

Options

A. `2 veca. vecb= |vecb|^(2)`
B. ` veca. vecb= |vecb|^(2)`
C. least value of `veca . Vecb + 1/(|vecb|^(2) + 2) " is " sqrt2`
D. least value of `veca .vecb + 1/(|vecb|^(2) + 2) " is " sqrt2 -1 `

Explanation

To solve the problem, we start with the given equation involving the magnitudes of the vectors:

Given:
\[ |\vec{a} + \vec{b}| = |\vec{a} - 2\vec{b}| \]

### Step 1: Square both sides
To eliminate the absolute values, we square both sides:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a} - 2\vec{b}|^2 \]

### Step 2: Expand both sides
Using the property that \(|\vec{u}|^2 = \vec{u} \cdot \vec{u}\), we expand both sides:
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (\vec{a} - 2\vec{b}) \cdot (\vec{a} - 2\vec{b}) \]

This gives us:
\[ \vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \vec{a} \cdot \vec{a} - 4\vec{a} \cdot \vec{b} + 4\vec{b} \cdot \vec{b} \]

### Step 3: Simplify the equation
Now we simplify both sides:
\[ |\vec{a}|^2 + 2\vec{a} \cdot \vec{b} + |\vec{b}|^2 = |\vec{a}|^2 - 4\vec{a} \cdot \vec{b} + 4|\vec{b}|^2 \]

### Step 4: Rearrange the equation
Subtract \(|\vec{a}|^2\) from both sides:
\[ 2\vec{a} \cdot \vec{b} + |\vec{b}|^2 = -4\vec{a} \cdot \vec{b} + 4|\vec{b}|^2 \]

Combine like terms:
\[ 2\vec{a} \cdot \vec{b} + 4\vec{a} \cdot \vec{b} = 4|\vec{b}|^2 - |\vec{b}|^2 \]
\[ 6\vec{a} \cdot \vec{b} = 3|\vec{b}|^2 \]

### Step 5: Solve for \( \vec{a} \cdot \vec{b} \)
Dividing both sides by 3:
\[ 2\vec{a} \cdot \vec{b} = |\vec{b}|^2 \]

### Conclusion
Thus, we have:
\[ \vec{a} \cdot \vec{b} = \frac{1}{2} |\vec{b}|^2 \]

### Final Answer
The correct option based on our calculations is:
- Option A: \(2\vec{a} \cdot \vec{b} = |\vec{b}|^2\)