If `alpha,beta,gamma` satisfy `k xx(kxxa) =0 and a =alphai+betaj+gammak`, where `alpha +beta+gamma=2`, then `gamma =` ........

Answer

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

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

### Step 2: Apply the Vector Triple Product Identity
Using the vector triple product identity:
\[ \hat{k} \times (\hat{k} \times \mathbf{a}) = (\hat{k} \cdot \mathbf{a}) \hat{k} - (\hat{k} \cdot \hat{k}) \mathbf{a} \]
Since \( \hat{k} \cdot \hat{k} = 1 \), we can rewrite the equation as:
\[ \hat{k} \times (\hat{k} \times \mathbf{a}) = (\hat{k} \cdot \mathbf{a}) \hat{k} - \mathbf{a} \]

### Step 3: Calculate \( \hat{k} \cdot \mathbf{a} \)
Now, we need to compute \( \hat{k} \cdot \mathbf{a} \):
\[ \hat{k} \cdot \mathbf{a} = \hat{k} \cdot (\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}) = 0 + 0 + \gamma = \gamma \]
Thus, we have:
\[ \hat{k} \times (\hat{k} \times \mathbf{a}) = \gamma \hat{k} - \mathbf{a} \]

### Step 4: Set the Equation to Zero
Since we know that \( \hat{k} \times (\hat{k} \times \mathbf{a}) = 0 \), we set the equation to zero:
\[ \gamma \hat{k} - (\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}) = 0 \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ \gamma \hat{k} - \alpha \hat{i} - \beta \hat{j} - \gamma \hat{k} = 0 \]
This simplifies to:
\[ -\alpha \hat{i} - \beta \hat{j} = 0 \]

### Step 6: Comparing Coefficients
From the equation \( -\alpha \hat{i} - \beta \hat{j} = 0 \), we can conclude:
\[ \alpha = 0 \quad \text{and} \quad \beta = 0 \]

### Step 7: Substitute Back to Find \( \gamma \)
Now, substituting \( \alpha = 0 \) and \( \beta = 0 \) into the equation \( \alpha + \beta + \gamma = 2 \):
\[ 0 + 0 + \gamma = 2 \implies \gamma = 2 \]

### Final Answer
Thus, the value of \( \gamma \) is:
\[ \gamma = 2 \]

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