Question

If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)

Hint:

When vectors sum to zero, use vector properties to solve for angles between the vectors.

Answer 1

The Correct Option is C

Given the vector identity \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), it follows that \[\overrightarrow{a} = -(\overrightarrow{b} + \overrightarrow{c}).\] The angle \( \theta \) between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is determined by the dot product as follows: \[|\overrightarrow{b}| |\overrightarrow{c}| \cos \theta = \overrightarrow{b} \cdot \overrightarrow{c}.\] The value of \( \theta \) is \( \frac{\pi}{3} \).