Suppose a and b are two unit vectors and `theta` is acute angle between them. If `abs(a-b)^(2)=4sin^(2)(alphatheta)`, then `8alpha^(2)=` _______

Answer

To solve the problem step by step, we will start from the given equation and manipulate it to find the required value of \(8\alpha^2\).

### Step 1: Understand the given equation
We are given that:
\[ |a - b|^2 = 4 \sin^2(\alpha \theta) \]
where \(a\) and \(b\) are unit vectors.

### Step 2: Expand the left-hand side
Using the property of dot products, we can express \( |a - b|^2 \) as:
\[ |a - b|^2 = (a - b) \cdot (a - b) = a \cdot a - 2a \cdot b + b \cdot b \]
Since \(a\) and \(b\) are unit vectors, we have:
\[ a \cdot a = 1 \quad \text{and} \quad b \cdot b = 1 \]
Thus, we can simplify:
\[ |a - b|^2 = 1 - 2a \cdot b + 1 = 2 - 2a \cdot b \]

### Step 3: Set the two expressions equal
Now we equate the two expressions:
\[ 2 - 2a \cdot b = 4 \sin^2(\alpha \theta) \]

### Step 4: Isolate \(a \cdot b\)
Rearranging gives:
\[ 2 - 4 \sin^2(\alpha \theta) = 2a \cdot b \]
Dividing the entire equation by 2:
\[ 1 - 2 \sin^2(\alpha \theta) = a \cdot b \]

### Step 5: Use the cosine identity
Recall the identity \( \cos \theta = a \cdot b \). Therefore, we can write:
\[ \cos \theta = 1 - 2 \sin^2(\alpha \theta) \]
Using the identity \(1 - 2 \sin^2 x = \cos 2x\), we can rewrite this as:
\[ \cos \theta = \cos(2\alpha \theta) \]

### Step 6: Set the angles equal
Since both \(\theta\) and \(2\alpha \theta\) are angles where the cosine values are equal, we can conclude:
\[ \theta = 2\alpha \theta \]
Assuming \(\theta \neq 0\), we can divide both sides by \(\theta\):
\[ 1 = 2\alpha \implies \alpha = \frac{1}{2} \]

### Step 7: Calculate \(8\alpha^2\)
Now we need to find \(8\alpha^2\):
\[ 8\alpha^2 = 8\left(\frac{1}{2}\right)^2 = 8 \cdot \frac{1}{4} = 2 \]

### Final Answer
Thus, the value of \(8\alpha^2\) is:
\[ \boxed{2} \]