If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three vectors such that \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), \(|\vec{a}| = 7\), \(|\vec{b}| = 3\) and \(|\vec{c}| = 5\), then angle between \(\vec{b}\) and \(\vec{c}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{\pi}{3}\)
Since \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), we have \(\vec{b} + \vec{c} = -\vec{a}\). Squaring the magnitudes, with \(\vec{b} \cdot \vec{c} = |\vec{b}||\vec{c}|\cos\theta\):
\(|\vec{b} + \vec{c}|^2 = |\vec{a}|^2\)
\(|\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{b}||\vec{c}|\cos\theta = |\vec{a}|^2\)
\(9 + 25 + 30\cos\theta = 49\)
\(30\cos\theta = 15\)
\( \displaystyle \cos\theta = \frac{1}{2},\quad \theta = \frac{\pi}{3} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{2\pi}{3}\) — Incorrect. This is the interior angle of the triangle with sides 3, 5, 7 formed by the three vectors placed head to tail. The angle between \(\vec{b}\) and \(\vec{c}\) themselves is its supplement, \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \).
Option (2) \(\displaystyle\frac{\pi}{6}\) — Incorrect. Its cosine is \( \frac{\sqrt{3}}{2} \), not \( \frac{1}{2} \).
Option (3) \(\displaystyle\frac{5\pi}{6}\) — Incorrect. Its cosine is \( -\frac{\sqrt{3}}{2} \), not \( \frac{1}{2} \).