Question

Let $\vec{a}=2\hat{i}-7\hat{j}+5\hat{k},\vec{b}=\hat{i}+\hat{k}$ and $\vec{c}=\hat{i}+2\hat{j}-3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=\vec{c}\times \vec{a}$ and $\vec{r}\cdot \vec{b}=0$, then $|\vec{r}|$ is equal to :

• A. $\frac{11}{7}\sqrt{2}$
• B. $\frac{11}{7}$
• C. $\frac{\sqrt{914}}{7}$
• D. $\frac{11}{5}\sqrt{2}$

Answer 1

Answer: (A)

$\vec{a}=2\hat{i}-7\hat{j}+5\hat{k}$
$\vec{b}=\hat{i}+\hat{k}$
$\vec{c}=\hat{i}+2\hat{j}-3\hat{k}$
$\vec{r}\times \vec{a}=\vec{c}\times \vec{a}\Rightarrow (\vec{r}-\vec{c})\times \vec{a}=0$
$\therefore \vec{r}=\vec{c}+\lambda \vec{a}$
$\vec{r}\cdot \vec{b}=0\Rightarrow \vec{c}\cdot \vec{b}+\lambda \vec{b}\cdot \vec{a}=0$
$-2+\lambda (7)=0\Rightarrow \lambda =\frac{2}{7}$
$\therefore \vec{r}=\vec{c}+\frac{2\vec{a}}{7}=\frac{1}{7}(11\hat{i}-11\hat{k})$
$\mid \vec{r}=\frac{11\sqrt{2}}{7}$