\[ \frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}} = ? \] Let \( \theta \) be the angle between the vectors \( \vec{a} \) and \( \vec{b} \), where \( 0 < \theta < \pi \).
Step 1: Use the formulas:
Magnitude of cross product: \[ |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta \] Dot product: \[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta \] Step 2: Plug into the expression: \[ \frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}} = \frac{|\vec{a}||\vec{b}|\sin\theta}{|\vec{a}||\vec{b}|\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta \] Final Answer: \( {\tan\theta} \)