Question

Find out whether the following pair of lines are parallel, non-parallel & intersecting, or non-parallel & non-intersecting.
$\overrightarrow{r_1}=\hat{i}-\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+\hat{k})$
$\overrightarrow{r_2}=2\hat{i}+4\hat{j}+6\hat{k}+\mu (2\hat{i}+\hat{j}+3\hat{k})$.

Answer 1

Given line,

$$\begin{array}{c}\overrightarrow{r_1}=\hat{i}-\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+\hat{k})=\overline{a}+\lambda \overline{b}-------\left(1\right)\\ \overrightarrow{r_2}=2\hat{i}+4\hat{j}+6\hat{k}+\mu (2\hat{i}+\hat{j}+3\hat{k})=\overline{c}+\mu \overline{d}-------\left(2\right)\\ \frac{1}{2}\ne \frac{-1}{4}=\frac{1}{3}\end{array}$$ lines are not parallel

then,

$\begin{array}{c}\left[\overline{a}-\overline{c}\overline{b}\overline{d}\right]\\ \therefore \overline{a}-\overline{c}=(1-1+3)-(2+4+6)\\ =-1-5-3\\ \left[\begin{array}{c}153\\ 1-11\\ 213\end{array}\right]=1(-3-1)-5(3-2)+3(1+2)\\ =-4-5+9\\ =0\\ \therefore linearecoplanear.\\ Let,\overline{r}isPOIofgivenlines,\end{array}$

$\begin{array}{c}Now,fromtheequation\left(1\right)+\left(2\right)\\ \Rightarrow 1+\lambda =2+2\mu -----\left(3\right)\\ \Rightarrow -1-\lambda =4+\mu -----\left(4\right)\\ \Rightarrow 3+\lambda =6+3\mu ------\left(5\right)\\ thenfromequation\left(3\right)+\left(4\right)wegetthevalueof\mu and\lambda .\\ 2+2\mu +4+\mu =0\\ 6+3\mu =0\\ \therefore \mu =-2\end{array}$

$\begin{array}{c}fromequation\left(5\right)\\ 3+\lambda =6+3\mu \\ 3+\lambda =6+3(-2)\\ \therefore \lambda =-3\\ So,\\ thelinesarenon-parallelandnon-intersecting.\end{array}$