Question

$\overrightarrow{r_1}=\hat{i}+\hat{k}+\lambda (\hat{i}+3\hat{j}+4\hat{k})$
$\overrightarrow{r_2}=2\hat{i}+3\hat{j}+\mu (4\hat{i}-\hat{j}+\hat{k})$

Answer 1

The direction ratios of the two lines are not equal, so the lines are not parallel. if the lines intersect, it will be $r_1=r_2$. i.e., where,

$(1+\lambda )i+3\lambda j+(1+4\lambda )k=(2+4\mu )i+(3-\mu )j+\mu k$

Equating the coefficients of $i$ and $j$, we have $1+\lambda =2+4\mu$ and $3\lambda =3-\mu$

Hence, $\mu =0,\lambda =1$

With the value of $\lambda$ and $\mu$ coefficient of $k$ bcome

first line $1+4\lambda =5$

second line $\mu =0$

Unequal values

So there is no value of $\lambda$ and $\mu$ for which $r_1=r_2$ and the lines do not intersect and are skew.