Question

If \(|\overrightarrow{u}|=5,|\overrightarrow{v}|=4\) and the angle between \(\overrightarrow{u}\) and \(\overrightarrow{v}\) is \(\frac{\pi}{6}\), then \(|\overrightarrow{u}\times\overrightarrow{v}|\) is equal to

• A. 10√3
• B. 10√2
• C. 20
• D. 5√2
• E. 10

Answer 1

Answer: (E)

Step 1: Use the formula for the magnitude of the cross product  
The magnitude of the cross product of two vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \) is given by: \[ |\overrightarrow{u} \times \overrightarrow{v}| = |\overrightarrow{u}| |\overrightarrow{v}| \sin\theta \] Given: \[ |\overrightarrow{u}| = 5, \quad |\overrightarrow{v}| = 4, \quad \theta = \frac{\pi}{6} \]

Step 2: Compute the sine value 
\[ \sin\frac{\pi}{6} = \frac{1}{2} \]

Step 3: Calculate the magnitude 
\[ |\overrightarrow{u} \times \overrightarrow{v}| = (5)(4) \times \frac{1}{2} \] \[ = 20 \times \frac{1}{2} \] \[ = 10 \]

Final Answer: \( 10 \).