Question

If \( \vec{a}, \vec{b}, \vec{c} \) are 3 vectors such that \( |\vec{a}| = 5, |\vec{b}| = 8, |\vec{c}| = 11 \) and \( \vec{a} + \vec{b} + \vec{c} = 0 \), then the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is:

• A. \( \cos^{-1}\left( \frac{-2}{5} \right) \)
• B. \( \cos^{-1}\left( \frac{10}{11} \right) \)
• C. \( \cos^{-1}\left( \frac{41}{55} \right) \)
• D. \( \frac{\pi}{3} \)

Hint:

When vectors are in equilibrium (i.e., sum to zero), you can use their magnitudes and vector addition properties to find the angle between them.

Answer 1

The Correct Option is A

We are given the following conditions: \[ |\vec{a}| = 5, \quad |\vec{b}| = 8, \quad |\vec{c}| = 11, \quad \vec{a} + \vec{b} + \vec{c} = 0. \] Using the property \( \vec{a} + \vec{b} + \vec{c} = 0 \), we can write \( \vec{c} = -(\vec{a} + \vec{b}) \).
Now, to find the angle \( \theta \) between \( \vec{a} \) and \( \vec{b} \), we use the formula for the dot product: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta. \] Substitute the known values and solve for \( \cos \theta \), we find that: \[ \cos \theta = \frac{-2}{5}. \] Thus, the correct answer is \( \cos^{-1}\left( \frac{-2}{5} \right) \).