Question

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:

• A. 144
• B. \(\sqrt{12}\)
• C. 12
• D. 24

Answer 1

The Correct Option is C

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:

\[ \boxed{12}. \]