Let $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\vec{c} = |\vec{c}||\vec{a} - \vec{a}| = 2\sqrt{2}$ and the angle between $(\vec{a} \times \vec{b})$ and $\vec{c}$ is $30°$, then the value of $|(\vec{a} \times \vec{b}) \times 2\vec{c}|$ is _______.
Step-by-Step Solution
Key Concept: The scalar triple product identity relates cross product magnitudes to angles; use the constraint on $|\vec{c} - \vec{a}|$ to determine $|\vec{c}|$.
Computing $\vec{a} \times \vec{b} = 2\vec{i} - 2\vec{j} + \vec{k}$, so $|\vec{a} \times \vec{b}| = 3$. Using $|(\vec{a} \times \vec{b}) \times \vec{c}| = |\vec{a} \times \vec{b}| |\vec{c}| \sin 30°$, and given $|\vec{c} - \vec{a}|^2 = 8$, we expand to find $|\vec{c}| = 2$. Therefore $|(\vec{a} \times \vec{b}) \times \vec{c}| = 3 \cdot 2 \cdot \frac{1}{2} = \frac{3}{2}$.
Correct Answer: Looking at this problem, I need to find the final answer for $|(\vec{a} \times \vec{b}) \times 2\vec{c}|$.
Let me trace through the solution:
**Step 1: Calculate $\vec{a} \times \vec{b}$**
$$\vec{a} \times \vec{