If \(\vec a + \vec b + \vec c = \vec 0\) and \(|\vec a| = 3, |\vec b| = 5, |\vec c| = 7\), then the angle between \(\vec a\) and \(\vec b\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\pi}{3}\)
Since \(\vec a + \vec b = -\vec c\), squaring the magnitudes (\(|\vec v|^2 = \vec v \cdot \vec v\)):
\(|\vec a + \vec b|^2 = |\vec c|^2\)
\(|\vec a|^2 + |\vec b|^2 + 2\,\vec a \cdot \vec b = |\vec c|^2\)
\(9 + 25 + 2\,\vec a \cdot \vec b = 49\)
\(\displaystyle \vec a \cdot \vec b = \frac{15}{2}\)
By the definition of the scalar product, \(\vec a \cdot \vec b = |\vec a|\,|\vec b|\cos\theta\):
\(\displaystyle \cos\theta = \frac{15/2}{3 \times 5} = \frac{1}{2}\)
\(\displaystyle \theta = \frac{\pi}{3}\)