Question

$\hat{u}$ and $\hat{v}$ are unit vectors such that $\hat{u}.\hat{v}=0\overrightarrow{r}$ is any vector coplanar with $\hat{u}$ and $\hat{v}$ the magnitude of the vector $\overrightarrow{r}\times(\hat{u}\times\hat{v}$ is

• A. 0
• B. 1
• C. |$\overrightarrow{r}$|
• D. 2|$\overrightarrow{r}$|

Answer 1

Answer: (C)

|$\overrightarrow{r}$|

Answer 2

$\bar{r}\times\left(\bar{u}\times\bar{v}\right)=\left(\lambda\,\bar{u}\times\mu\,\bar{v}\right)\times\left(\bar{u} \times \bar{v}\right)$ $=\lambda \bar{\mu} \times \left(\bar{u} \times\bar{v}\right)+\mu\,\bar{v} \times \left(\bar{u} \times \bar{v}\right)$ $=\lambda \left[\left(\bar{u}\cdot\bar{v}\right)\bar{u}-\left(\bar{\mu}\cdot\bar{u}\right)\bar{v}\right]+\mu\left[\left(\bar{v}\cdot\bar{v}\right)\bar{u}-\left(\bar{u}\cdot\bar{v}\right)\bar{v}\right]$ $=\lambda\left(0-\left|\bar{u}\right|^{2}\,\bar{v}\right)+\mu\left(\left|\bar{v}\right|^{2}\,\bar{u}-0\right)$ $=-\lambda\,\bar{v}-r\,\bar{u}$ $\therefore \left|\bar{r}\times\left(\bar{u} \times\bar{v}\right)\right|=1r$