If `vec a,vec b,vec c` are three non-zero vectors and `hatn` is a unit vector perpendicular to `vec c` such that `vec a=alpha vec b-hatn,(alpha!=0)` and `vec b*vec c=12`,then `|vec c times(vec a times vec b)|` is equal to :

Options

A. `9`
B. `15`
C. `6`
D. `12`

Answer

To solve the problem, we need to find the value of \(|\vec{c} \times (\vec{a} \times \vec{b})|\) given the relationships between the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

### Step 1: Understand the given relationships
We have:
- \(\vec{a} = \alpha \vec{b} - \hat{n}\) where \(\hat{n}\) is a unit vector perpendicular to \(\vec{c}\).
- \(\vec{b} \cdot \vec{c} = 12\).

### Step 2: Find \(\hat{n} \cdot \vec{c}\)
Since \(\hat{n}\) is perpendicular to \(\vec{c}\), we have:
\[ \hat{n} \cdot \vec{c} = 0. \]

### Step 3: Calculate \(\vec{a} \cdot \vec{c}\)
Using the expression for \(\vec{a}\):
\[ \vec{a} \cdot \vec{c} = (\alpha \vec{b} - \hat{n}) \cdot \vec{c} = \alpha (\vec{b} \cdot \vec{c}) - (\hat{n} \cdot \vec{c}). \]
Substituting the known values:
\[ \vec{a} \cdot \vec{c} = \alpha (12) - 0 = 12\alpha. \]

### Step 4: Use the vector triple product identity
We need to compute \(|\vec{c} \times (\vec{a} \times \vec{b})|\). Using the vector triple product identity:
\[ \vec{c} \times (\vec{a} \times \vec{b}) = (\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b}. \]
Now substituting the values we have:
\[ |\vec{c} \times (\vec{a} \times \vec{b})| = |(\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b}|. \]
Substituting \(\vec{c} \cdot \vec{b} = 12\) and \(\vec{c} \cdot \vec{a} = 12\alpha\):
\[ |\vec{c} \times (\vec{a} \times \vec{b})| = |12 \vec{a} - 12\alpha \vec{b}|. \]

### Step 5: Factor out the common term
Factoring out 12:
\[ |\vec{c} \times (\vec{a} \times \vec{b})| = 12 |\vec{a} - \alpha \vec{b}|. \]

### Step 6: Substitute \(\vec{a}\)
We know \(\vec{a} = \alpha \vec{b} - \hat{n}\), so:
\[ |\vec{a} - \alpha \vec{b}| = |(\alpha \vec{b} - \hat{n}) - \alpha \vec{b}| = |-\hat{n}| = 1. \]

### Step 7: Final calculation
Thus,
\[ |\vec{c} \times (\vec{a} \times \vec{b})| = 12 \cdot 1 = 12. \]

### Conclusion
The value of \(|\vec{c} \times (\vec{a} \times \vec{b})|\) is \(\boxed{12}\).