Question
By computing the shortest distance determine whether following lines intersect eachother.
$\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-\hat{j}+\hat{k})$ and $\bar{r}=(2 \hat{i}+2 \hat{j}-3 \hat{k})+\mu(\hat{i}+\hat{j}-2 \hat{k})$
$\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-\hat{j}+\hat{k})$ and $\bar{r}=(2 \hat{i}+2 \hat{j}-3 \hat{k})+\mu(\hat{i}+\hat{j}-2 \hat{k})$
$\bar{r}=\bar{a}_1+\lambda \bar{b}_1$ and $\bar{r}=\bar{a}_2+\mu \bar{b}_2$ is given by
$d=\left|\frac{\left(\bar{a}_2-\bar{a}_1\right) \cdot\left(\bar{b}_1 \times \bar{b}_2\right)}{\left|\bar{b}_1 \times \bar{b}_2\right|}\right|$.
Here, $\bar{a}_1=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \bar{a}_2=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$,
$\begin{aligned} & \overline{\mathrm{b}}_1=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}_2=\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}} \\ & \therefore \overline{\mathrm{b}}_1 \times \overline{\mathrm{b}}_2=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 2 & -1 & 1 \\ 1 & 1 & -2\end{array}\right| \\ & =(2-1) \hat{\mathrm{i}}-(-4-1) \hat{j}+(4+1) \hat{k} \\ & =\hat{\mathrm{i}}-5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\end{aligned}$
$\begin{aligned} & \overline{\mathrm{a}}_2-\overline{\mathrm{a}}_1=(2 \hat{\mathrm{i}}+2 \hat{j}-3 \hat{k})-(\hat{i}+\hat{j}-\hat{k}) \\ & \therefore\left(\bar{a}_2-\overline{\mathrm{a}}_1\right) \cdot\left(\bar{b}_1 \times \bar{b}_2\right)=\hat{i} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k}) \\ & =1(-1)+0(3)+0(2) \\ & =-1\end{aligned}$
and
$\begin{aligned} & \left|\bar{b}_1 \times \bar{b}_2\right|=\sqrt{(-1)^2+3^2+2^2} \\ & =\sqrt{1+9+4} \\ & =\sqrt{14}\end{aligned}$
Shortest distance between the lines is 0. ∴ the lines intersect each other.
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$\left(1-x^2\right) \frac{d y}{d x}+2 x y=x\left(1-x^2\right)^{\frac{1}{2}}$