If \(\hat{a}, \hat{b}\) and \(\hat{c}\) are three unit vectors and \(\hat{a} + \hat{b} + \hat{c} = \vec{0}\), then the angle between \(\hat{a}\) and \((-\hat{b})\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\pi}{3}\)
From the given sum, \(\hat{a} + \hat{b} = -\hat{c}\). Taking the magnitude squared of both sides:
\(|\hat{a} + \hat{b}|^2 = |\hat{c}|^2 = 1\)
\(|\hat{a}|^2 + |\hat{b}|^2 + 2\,\hat{a} \cdot \hat{b} = 1\)
\(1 + 1 + 2\,\hat{a} \cdot \hat{b} = 1\)
\( \displaystyle \hat{a} \cdot \hat{b} = -\frac{1}{2} \)
If \(\theta\) is the angle between \(\hat{a}\) and \(-\hat{b}\), then by the dot product:
\( \displaystyle \cos\theta = \hat{a} \cdot (-\hat{b}) = -\hat{a} \cdot \hat{b} = \frac{1}{2} \)
\( \displaystyle \theta = \frac{\pi}{3} \)
Tip: The angle between \(\hat{a}\) and \(\hat{b}\) themselves is \(\frac{2\pi}{3}\) (Option 1). The question asks for \(\hat{a}\) and \(-\hat{b}\); reversing one vector changes the angle \(\alpha\) to \(\pi - \alpha\).