If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are three non-zero vectors and \(\hat{\mathbf{n}}\) is a unit vector perpendicular to \(\mathbf{c}\) such that: \(\mathbf{a} = \alpha \mathbf{b} - \hat{\mathbf{n}}, \, (\alpha \neq 0)\) and \( \mathbf{b} \cdot \mathbf{c} = 12 \), then \( \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) \) is equal to:
- 144
\(\sqrt{12}\)
- 12
- 24
The Correct Option is C
Solution and Explanation
We are given:
- \(\vec{n} \perp \vec{c}\),
- \(\vec{a} = \alpha \vec{b} - \vec{n}\),
- \(\vec{b} \cdot \vec{c} = 12\).
We need to find:
\[ |\vec{c} \times (\vec{a} \times \vec{b})|. \]
Step 1: Expand \(\vec{a} \times \vec{b}\)
Using the distributive property, expand:
\[ \vec{a} \times \vec{b} = (\alpha \vec{b} - \vec{n}) \times \vec{b}. \]
Distribute the terms:
\[ \vec{a} \times \vec{b} = \alpha (\vec{b} \times \vec{b}) - (\vec{n} \times \vec{b}). \]
Since \(\vec{b} \times \vec{b} = 0\), this simplifies to:
\[ \vec{a} \times \vec{b} = -(\vec{n} \times \vec{b}). \]
Step 2: Compute \(\vec{c} \times (\vec{a} \times \vec{b})\)
Substitute \(\vec{a} \times \vec{b}\) into \(\vec{c} \times (\vec{a} \times \vec{b})\):
\[ \vec{c} \times (\vec{a} \times \vec{b}) = \vec{c} \times (-\vec{n} \times \vec{b}). \]
Using the vector triple product identity:
\[ \vec{c} \times (\vec{n} \times \vec{b}) = (\vec{c} \cdot \vec{b}) \vec{n} - (\vec{c} \cdot \vec{n}) \vec{b}. \]
Substitute this back, considering the negative sign:
\[ \vec{c} \times (\vec{a} \times \vec{b}) = -((\vec{c} \cdot \vec{b}) \vec{n} - (\vec{c} \cdot \vec{n}) \vec{b}). \]
Given \(\vec{n} \perp \vec{c}\), we have \(\vec{c} \cdot \vec{n} = 0\). Substituting this value:
\[ \vec{c} \times (\vec{a} \times \vec{b}) = -(\vec{c} \cdot \vec{b}) \vec{n}. \]
Step 3: Magnitude of \(\vec{c} \times (\vec{a} \times \vec{b})\)
Given \(\vec{b} \cdot \vec{c} = 12\), substitute this value:
\[ \vec{c} \times (\vec{a} \times \vec{b}) = -12 \vec{n}. \]
Compute the magnitude:
\[ |\vec{c} \times (\vec{a} \times \vec{b})| = 12 |\vec{n}|. \]
Since \(\vec{n}\) is a unit vector, \(|\vec{n}| = 1\). Thus:
\[ |\vec{c} \times (\vec{a} \times \vec{b})| = 12. \]
Conclusion
The value of \(|\vec{c} \times (\vec{a} \times \vec{b})|\) is:
\[ \boxed{12}. \]
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