Question

If $ \theta $ is the angle between two vectors $ \vec{a} $ and $ \vec{b} $, then $ | \vec{a} \times \vec{b} | = | \vec{a} \cdot \vec{b} | $ equals to

• A. \( \cot \theta \)
• B. \( -\cot \theta \)
• C. \( \tan \theta \)
• D. \( -\tan \theta \)

Hint:

For the magnitudes of cross and dot products, use the relations \( | \vec{a} \times \vec{b} | = |\vec{a}| |\vec{b}| \sin \theta \) and \( | \vec{a} \cdot \vec{b} | = |\vec{a}| |\vec{b}| \cos \theta \).

Answer 1

The Correct Option is C

Step 1: Recall the Formulas for Cross and Dot Products The magnitude of the cross product is given by: \[ | \vec{a} \times \vec{b} | = |\vec{a}| |\vec{b}| \sin \theta \] The magnitude of the dot product is: \[ | \vec{a} \cdot \vec{b} | = |\vec{a}| |\vec{b}| \cos \theta \]
Step 2: Relate the Two Magnitudes We are asked to find the relationship between the magnitudes of the cross product and the dot product.
Dividing the cross product magnitude by the dot product magnitude, we get: \[ \frac{| \vec{a} \times \vec{b} |}{| \vec{a} \cdot \vec{b} |} = \frac{|\vec{a}| |\vec{b}| \sin \theta}{|\vec{a}| |\vec{b}| \cos \theta} = \tan \theta \]
Step 3: Conclusion Thus, the correct answer is \( \tan \theta \).