If θ is the angle between two unit vectors a and b then |a - b| =

The correct answer is Option (2) → \(2\sin\displaystyle\frac{\theta}{2}\)

Since \(|\hat{a}| = |\hat{b}| = 1\) and \(\hat{a}\cdot\hat{b} = \cos\theta\), the square of the magnitude is

    \(|\hat{a} - \hat{b}|^2 = (\hat{a} - \hat{b})\cdot(\hat{a} - \hat{b})\)
    \(= |\hat{a}|^2 + |\hat{b}|^2 - 2\,\hat{a}\cdot\hat{b}\)
    \(= 2 - 2\cos\theta\)

Using the half-angle identity \(1 - \cos\theta = 2\sin^2\frac{\theta}{2}\):

    \(\displaystyle |\hat{a} - \hat{b}|^2 = 4\sin^2\frac{\theta}{2}\)

Since \(0 \le \theta \le \pi\), \(\sin\frac{\theta}{2} \ge 0\), so taking the positive square root:

    \(\displaystyle |\hat{a} - \hat{b}| = 2\sin\frac{\theta}{2}\)

Tip: Remember the pair \(|\hat{a} - \hat{b}| = 2\sin\frac{\theta}{2}\) and \(|\hat{a} + \hat{b}| = 2\cos\frac{\theta}{2}\). The second one is Option (4), an easy mix-up.