Key Concepts and Formulas

• Vector Triple Product Identity: For any three vectors $\vec{x}, \vec{y}, \vec{z}$, the vector triple product is given by: $\vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z})\vec{y} - (\vec{x} \cdot \vec{y})\vec{z}$
• Properties of Dot Product:
  • Commutative: $\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}$
  • Distributive: $\vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w}$
  • Scalar multiplication: $\vec{u} \cdot (k\vec{v}) = k(\vec{u} \cdot \vec{v})$
• Perpendicular Vectors: If vector $\vec{u}$ is perpendicular to vector $\vec{v}$, then $\vec{u} \cdot \vec{v} = 0$.
• Unit Vector: A unit vector $\widehat{n}$ has a magnitude of 1, i.e., $|\widehat{n}| = 1$.

Step-by-Step Solution

Step 1: Apply the Vector Triple Product Formula We need to find the magnitude of $\vec{c} \times (\vec{a} \times \vec{b})$. We begin by simplifying the vector triple product expression using the identity. Let $\vec{x} = \vec{c}$, $\vec{y} = \vec{a}$, and $\vec{z} = \vec{b}$. $\vec{c} \times (\vec{a} \times \vec{b}) = (\vec{c} \cdot \vec{b})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ Explanation: This formula is essential for resolving the vector triple product into a linear combination of the constituent vectors, making it easier to work with.

Step 2: Substitute Known Dot Product Values We are given that $\vec{b} \cdot \vec{c} = 12$. Since the dot product is commutative, $\vec{c} \cdot \vec{b} = 12$. Substitute this into the equation from Step 1. $\vec{c} \times (\vec{a} \times \vec{b}) = (12)\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ $\vec{c} \times (\vec{a} \times \vec{b}) = 12\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ Explanation: This step directly incorporates one of the given numerical values, simplifying the first term of the resulting vector expression.

Step 3: Evaluate the Dot Product $\vec{c} \cdot \vec{a}$ We are given the relationship $\vec{a} = \alpha \vec{b} - \widehat{n}$. We can find $\vec{c} \cdot \vec{a}$ by taking the dot product of $\vec{c}$ with this expression for $\vec{a}$. $\vec{c} \cdot \vec{a} = \vec{c} \cdot (\alpha \vec{b} - \widehat{n})$ Using the distributive and scalar multiplication properties of the dot product: $\vec{c} \cdot \vec{a} = \alpha (\vec{c} \cdot \vec{b}) - (\vec{c} \cdot \widehat{n})$ We are given $\vec{c} \cdot \vec{b} = 12$. We are also told that $\widehat{n}$ is perpendicular to $\vec{c}$, which means $\vec{c} \cdot \widehat{n} = 0$. Substituting these values: $\vec{c} \cdot \vec{a} = \alpha (12) - 0$ $\vec{c} \cdot \vec{a} = 12\alpha$ Explanation: This step is crucial for finding the coefficient of $\vec{b}$ in the simplified expression. It leverages the given vector relation and the property that $\widehat{n}$ is perpendicular to $\vec{c}$.

Step 4: Substitute $\vec{c} \cdot \vec{a}$ back into the Expression Now, substitute the value of $\vec{c} \cdot \vec{a} = 12\alpha$ back into the equation from Step 2. $\vec{c} \times (\vec{a} \times \vec{b}) = 12\vec{a} - (12\alpha)\vec{b}$ We can factor out 12 from both terms: $\vec{c} \times (\vec{a} \times \vec{b}) = 12(\vec{a} - \alpha\vec{b})$ Explanation: This step consolidates the expression for the vector triple product, bringing us closer to evaluating its magnitude.

Step 5: Use the Given Relationship for $\vec{a}$ Recall the given relationship: $\vec{a} = \alpha \vec{b} - \widehat{n}$. Rearranging this equation, we get $\vec{a} - \alpha\vec{b} = -\widehat{n}$. Substitute this into the expression from Step 4: $\vec{c} \times (\vec{a} \times \vec{b}) = 12(-\widehat{n})$ $\vec{c} \times (\vec{a} \times \vec{b}) = -12\widehat{n}$ Explanation: By substituting the given relation for $\vec{a}$, we express the entire vector triple product in terms of the unit vector $\widehat{n}$.

Step 6: Calculate the Magnitude We need to find the magnitude of the resulting vector: $|\vec{c} \times (\vec{a} \times \vec{b})|$. $\left| \vec{c} \times (\vec{a} \times \vec{b}) \right| = \left| -12\widehat{n} \right|$ Using the property $|k\vec{v}| = |k||\vec{v}|$: $\left| -12\widehat{n} \right| = |-12| |\widehat{n}|$ We know that $\widehat{n}$ is a unit vector, so $|\widehat{n}| = 1$. $|-12| |\widehat{n}| = 12 \times 1$ $\left| \vec{c} \times (\vec{a} \times \vec{b}) \right| = 12$ Explanation: The final step involves calculating the magnitude of the vector. Since $\widehat{n}$ is a unit vector, its magnitude is 1, which simplifies the calculation significantly.

Common Mistakes & Tips

• Incorrect Vector Triple Product Formula: Ensure you use the correct formula for the vector triple product. The order of vectors and dot products matters.
• Misinterpreting Perpendicularity: Remember that if $\widehat{n}$ is perpendicular to $\vec{c}$, their dot product $\widehat{n} \cdot \vec{c}$ (or $\vec{c} \cdot \widehat{n}$) is zero.
• Magnitude of Unit Vector: Always recall that a unit vector has a magnitude of 1. This is a common simplification.

Summary

The problem requires the application of the vector triple product identity to simplify the expression $\vec{c} \times (\vec{a} \times \vec{b})$. By systematically substituting the given conditions, particularly the relationship between $\vec{a}$ and $\vec{b}$ involving the unit vector $\widehat{n}$, and the values of the dot products, we were able to reduce the expression to a scalar multiple of $\widehat{n}$. The magnitude of this vector was then easily computed using the fact that $\widehat{n}$ is a unit vector.

The final answer is \boxed{12}.