Q73 / 160 · AP EAMCET 2021 (23 Aug Shift 2) · Mathematics

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

  • A
    $0$
  • B
    $1$
  • Correct answer
    C
    $|\mathbf{r}|$
  • D
    $2|\mathbf{r}|$

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 $\mathbf{r}$. $\therefore|\mathbf{r} \times(\mathbf{u} \times \mathbf{v})|=|\mathbf{r} \times \hat{\mathbf{n}}|=|| \mathbf{r}|| \hat{\mathbf{n}}\left|\sin \frac{\pi}{2}\right|=|\mathbf{r}|$

Hence, the correct answer is option: 'C'.

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