Let $\mathbf{u}$ and $\mathbf{v}$ are unit vectors such that $\mathbf{u} \cdot \mathbf{v}=0$. If…
Let $\mathbf{u}$ and $\mathbf{v}$ are unit vectors such that $\mathbf{u} \cdot \mathbf{v}=0$. If $\mathbf{r}$ is any vector coplanar with $\mathbf{u}$ and $\mathbf{v}$, then the magnitude of the vector $\mathbf{r} \times(\mathbf{u} \times \mathbf{v})$ is
$0$
$1$
$|\mathbf{r}|$
$2|\mathbf{r}|$
Solution
Given, $|\mathbf{u}|=|\mathbf{v}|=1, \mathbf{u} \cdot \mathbf{v}=0$
To find, $|\mathbf{r} \times(\mathbf{u} \times \mathbf{v})|$
$\because \quad \mathbf{u} \cdot \mathbf{v}=0$
$\Rightarrow|\mathbf{u}||\mathbf{v}| \cos \theta=0$
$\Rightarrow \quad \cos \theta=0$
$\theta=\frac{\pi}{2}$
Consider $\mathbf{u} \times \mathbf{v}=|\mathbf{u}||\mathbf{v}| \sin \frac{\pi}{2} \hat{\mathbf{n}}=\hat{\mathbf{n}}$
Since $\mathbf{r}$ is coplanar with $\mathbf{u}$ and $\mathbf{v}$.
Hence, $\hat{\mathbf{n}}$ is perpendicular to $\mathbf{r}$.
$\therefore|\mathbf{r} \times(\mathbf{u} \times \mathbf{v})|=|\mathbf{r} \times \hat{\mathbf{n}}|=|| \mathbf{r}|| \hat{\mathbf{n}}\left|\sin \frac{\pi}{2}\right|=|\mathbf{r}|$