$\vec{r}$ and $\vec{s}$ are unit vectors. If $|\vec{r} + \vec{s}| = \sqrt{2}$, find the angle between $\vec{r}$ and $\vec{s}$.

$\frac{\pi}{2}$

The correct answer is Option (3) → $\frac{\pi}{2}$ ##

Given, $|\vec{r}| = 1, |\vec{s}| = 1$ and $|\vec{r} + \vec{s}| = \sqrt{2}$ 

Now, $|\vec{r} + \vec{s}|^2 = (\sqrt{2})^2$

$\Rightarrow |\vec{r}|^2 + |\vec{s}|^2 + 2\vec{r} \cdot \vec{s} = 2$

$\Rightarrow 1^2 + 1^2 + 2\vec{r} \cdot \vec{s} = 2$

$\Rightarrow 2\vec{r} \cdot \vec{s} = 0$

$\Rightarrow \vec{r} \cdot \vec{s} = 0$

$\Rightarrow |\vec{r}||\vec{s}|\cos \theta = 0$

$\Rightarrow (1) \cdot (1) \cdot \cos \theta = 0$

$\Rightarrow \cos \theta = \cos \frac{\pi}{2}$

$\Rightarrow \theta = \frac{\pi}{2}$