Question

Let $\overrightarrow{a}=3\hat{i}+2\hat{j},\overrightarrow{b}=2\hat{i}-\hat{k},$ then the point of intersection of the lines $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$ is

• A. $5\hat{i}-2\hat{j}-\hat{k}$
• B. $5\hat{i}+2\hat{j}-\hat{k}$
• C. $-\hat{i}-2\hat{j}-\hat{k}$
• D. None of these

Answer 1

The correct option is B $5\hat{i}+2\hat{j}-\hat{k}$

Given, $\overline{r}\times \overline{a}=\overline{b}\times \overline{a}$

$\therefore (\overline{r}-\overline{b})\times \overline{a}=0$

The equation of the line is

$\overline{r}-\overline{b}=t\overline{a}$

i.e. $\overline{r}=\overline{b}+t\overline{a}$, where t is a scalar

Similarly, the equation of the other line is $\overline{r}=\overline{a}+s\overline{b}$, where $s$ is a scalar

If two lines intersect then,

$\overline{b}+t\overline{a}=\overline{a}+s\overline{b}$

$i.e.t=1$ and $s=1$

Point of intersection is $\overline{r}=\overline{a}+\overline{b}$,

i.e.$5i+2j-k$

Hence the answer is option $B$