If |a - r| = |a| = |r| = 1 , then angle between a and r is

The correct answer is Option (1) → \(\displaystyle\frac{\pi}{3}\)

Let \(\theta\) be the angle between \(\vec{a}\) and \(\vec{r}\), so that \(\vec{a}\cdot\vec{r} = |\vec{a}||\vec{r}|\cos\theta = \cos\theta\). Squaring \(|\vec{a} - \vec{r}| = 1\):

    \(|\vec{a} - \vec{r}|^2 = |\vec{a}|^2 + |\vec{r}|^2 - 2\,\vec{a}\cdot\vec{r}\)
    \(1 = 1 + 1 - 2\cos\theta\)

    \(\displaystyle \cos\theta = \frac{1}{2}\)

    \(\displaystyle \theta = \frac{\pi}{3}\)

Shortcut: \(\vec{a}\), \(\vec{r}\) and \(\vec{a} - \vec{r}\) form a triangle whose three sides all have length 1. It is an equilateral triangle, so the angle between \(\vec{a}\) and \(\vec{r}\) is \(\frac{\pi}{3}\).