If `alpha+beta+gamma =2 and veca=alphahati+betahatj+gammahatk, hatkxx (hatkxxveca)=vec0` then gamma= (A) 1 (B) -1 (C) 2 (D) none of these

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the problem step by step, we start with the given information and equations.

### Step 1: Understand the given equations
We are given that:
1. \( \alpha + \beta + \gamma = 2 \)
2. The vector \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \)
3. The equation \( \hat{k} \times (\hat{k} \times \vec{a}) = \vec{0} \)

### Step 2: Evaluate the cross product
Using the vector triple product identity, we can simplify \( \hat{k} \times (\hat{k} \times \vec{a}) \):
\[ \hat{k} \times (\hat{k} \times \vec{a}) = (\hat{k} \cdot \vec{a}) \hat{k} - (\hat{k} \cdot \hat{k}) \vec{a} \]
Since \( \hat{k} \cdot \hat{k} = 1 \), we can rewrite it as:
\[ \hat{k} \times (\hat{k} \times \vec{a}) = (\hat{k} \cdot \vec{a}) \hat{k} - \vec{a} \]

### Step 3: Calculate \( \hat{k} \cdot \vec{a} \)
The dot product \( \hat{k} \cdot \vec{a} \) is simply the k-component of \( \vec{a} \), which is \( \gamma \):
\[ \hat{k} \cdot \vec{a} = \gamma \]
Thus, we can substitute this back into our equation:
\[ \hat{k} \times (\hat{k} \times \vec{a}) = \gamma \hat{k} - \vec{a} \]

### Step 4: Set the equation to zero
Since we know that \( \hat{k} \times (\hat{k} \times \vec{a}) = \vec{0} \), we can set the equation:
\[ \gamma \hat{k} - \vec{a} = \vec{0} \]
This implies:
\[ \gamma \hat{k} = \vec{a} \]

### Step 5: Substitute \( \vec{a} \)
Substituting \( \vec{a} \) into the equation gives us:
\[ \gamma \hat{k} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \]

### Step 6: Equate components
From the equation \( \gamma \hat{k} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \), we can equate the components:
- The i-component: \( \alpha = 0 \)
- The j-component: \( \beta = 0 \)
- The k-component: \( \gamma = \gamma \)

### Step 7: Substitute back into the first equation
Now, substituting \( \alpha = 0 \) and \( \beta = 0 \) into the equation \( \alpha + \beta + \gamma = 2 \):
\[ 0 + 0 + \gamma = 2 \]
This simplifies to:
\[ \gamma = 2 \]

### Conclusion
Thus, the value of \( \gamma \) is \( 2 \), which corresponds to option (C).