If angle between \(\underline{a}\) and \(\underline{b}\) is \(\theta\), then \([(|\underline{\mathbf{a}} \times \underline{\mathrm{b}}|) /(\underline{\mathrm{a}} \cdot \underline{\mathrm{b}})]=\) \(\begin{array}{llll}(\text { A })-\cot \theta & \text { (B) }-\tan \theta & \text { (C) } \tan \theta & \text { (D) } \cot \theta\end{array}\)

We will use the following properties of cross and dot products: 1. \(|\underline{\mathbf{a}} \times \underline{\mathrm{b}}|=|\underline{\mathrm{a}}||\underline{\mathrm{b}}|\sin{\theta}\), where \(|\underline{\mathrm{a}}|\) and \(|\underline{\mathrm{b}}|\) are the magnitudes of vectors \(\underline{a}\) and \(\underline{b}\), respectively, and \(\theta\) is the angle between them. 2. \(\underline{\mathrm{a}} \cdot \underline{\mathrm{b}}=|\underline{\mathrm{a}}||\underline{\mathrm{b}}|\cos{\theta}\), where \(|\underline{\mathrm{a}}|\) and \(|\underline{\mathrm{b}}|\) are the magnitudes of vectors \(\underline{a}\) and \(\underline{b}\), respectively, and \(\theta\) is the angle between them. Now, we need to find the value of \([(|\underline{\mathbf{a}} \times \underline{\mathrm{b}}|)/(\underline{\mathrm{a}} \cdot \underline{\mathrm{b}})]\) using these properties.

We will substitute the expressions from step 1 into the requested expression: \(\frac{|\underline{\mathbf{a}} \times \underline{\mathrm{b}}| /(\underline{\mathrm{a}} \cdot \underline{\mathrm{b}})} = \frac{|\underline{\mathrm{a}}||\underline{\mathrm{b}}|\sin{\theta}}{|\underline{\mathrm{a}}||\underline{\mathrm{b}}|\cos{\theta}}\)

Cancel out the magnitudes of the two vectors and simplify: \(\frac{|\underline{\mathrm{a}}||\underline{\mathrm{b}}|\sin{\theta}}{|\underline{\mathrm{a}}||\underline{\mathrm{b}}|\cos{\theta}} = \frac{\sin{\theta}}{\cos{\theta}}\)

The expression \(\frac{\sin{\theta}}{\cos{\theta}}\) is equal to the tangent of the angle \(\theta\). So, we have: \(\frac{|\underline{\mathbf{a}} \times \underline{\mathrm{b}}| /(\underline{\mathrm{a}} \cdot \underline{\mathrm{b}})} = \tan{\theta}\) Thus, the correct answer is: (C) \(\tan \theta\).