Question

Assertion: The pair of lines given by $\vec{r}=\hat{i}-\hat{j}+\lambda (2i+k)$ and $\vec{r}=2\hat{i}-\hat{k}+\mu (i+\hat{j}-k)$ intersect.
Reason: Two lines intersect each other, if they are not parallel and shortest distance $=0$.

• A. Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
• B. Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
• C. Assertion is correct, Reason is incorrect
• D. Assertion is incorrect, Reason is correct.

Answer 1

Answer: (A)

Here, ${\vec{a}}_1=\hat{i}-\hat{j},{\vec{b}}_1=2\hat{i}+\hat{k}$
${\vec{a}}_2=2\hat{i}-\hat{k},{\vec{b}}_2=\hat{i}+\hat{j}-\hat{k}$
$\because {\vec{b}}_1\ne \lambda {\vec{b}}_2$, for any scalar $\lambda$
$\therefore$ Given lines are not parallel.
${\vec{a}}_2-{\vec{a}}_1=(2\hat{i}-\hat{k})-(\hat{i}-\hat{j})=\hat{i}+\hat{j}-\hat{k}$
${\vec{b}}_1\times {\vec{b}}_2=\left|\begin{array}{ccc}<br/><br/>\hat{i}& \hat{j}& \hat{k}\\ <br/><br/>2& 0& 1\\ <br/><br/>1& 1& -1<br/><br/>\end{array}\right|$
$=\hat{i}(0-1)-\hat{j}(-2-1)+\hat{k}(2-0)$
$=-\hat{i}+3\hat{j}+2\hat{k}$
$\left|{\vec{b}}_1\times {\vec{b}}_2\right|=\sqrt{(-1{)}^2+(3{)}^2+(2{)}^2}$
$=\sqrt{1+9+4}=\sqrt{14}$
$SD=\frac{\left({\vec{a}}_2-{\vec{a}}_1\right)\cdot \left({\vec{b}}_2-{\vec{b}}_1\right)}{|{\vec{b}}_1\times {\vec{b}}_2|}|$
$=\left|\frac{(\hat{i}+j-\hat{k})\cdot (-\hat{i}+3j+2\hat{k})}{\sqrt{14}}\right|$
$=\left|\frac{-1+3-2}{\sqrt{14}}\right|=0$
Hence, two lines intersect each other.
Two lines intersect each other,
if they are not parallel and shortest distance $=0$.