Question
By computing the shortest distance, determine whether following lines intersect each other.
$\bar{r}=(\hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{k})$ and $\bar{r}=(2 \hat{i}-\hat{j})+\mu(\hat{i}+\hat{j}-\hat{k})$
$\bar{r}=(\hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{k})$ and $\bar{r}=(2 \hat{i}-\hat{j})+\mu(\hat{i}+\hat{j}-\hat{k})$
The shortest distance between the lines
$\bar{r}=\overline{a_1}+\lambda \overline{b_1}$ and $\bar{r}=\overline{a_2}+\mu \overline{b_2}$ is given by
$d=\left|\frac{\left(\overline{a_2}-\overline{a_1}\right) \cdot\left(\overline{b_1} \times \overline{b_2}\right)}{\left|\overline{b_1} \times \overline{b_2}\right|}\right|$
Here, $\overline{a_1}=\hat{i}-\hat{j}, \overline{a_2}=2 \hat{i}-\hat{j}, \overline{b_1}=2 \hat{i}+\hat{k}, \overline{b_2}=\hat{i}+\hat{j}-\hat{k}$.
$\therefore \overline{b_1} \times \overline{b_2}=\left|\begin{array}{rrr}\hat{i} & \hat{j} & \hat{k} \\ 2 & 0 & 1 \\ 1 & 1 & -1\end{array}\right|$
$=(0-1) \hat{i}-(-2-1) \hat{j}+(2-0) \hat{k}$
$=-\vec{i}+3 \hat{j}+2 \hat{k}$
and $\overline{a_2}-\overline{a_1}=(2 \hat{i}-\hat{j})-(\hat{i}-\hat{j})=\hat{i}$
$\therefore\left(\overline{a_2}-\overline{a_1}\right) \cdot\left(\overline{b_1} \times \overline{b_2}\right)=\hat{i} \cdot(-\hat{i}+3 \hat{j}+2 \hat{k})$
$=1(-1)+0(3)+0(2)=-1$
and $\left|\overline{b_1} \times \overline{b_2}\right|=\sqrt{(-1)^2+3^2+2^2}$
$=\sqrt{1+9+4}=\sqrt{14}$
$\therefore$ the shortest distance between the given lines
$=\left|\frac{-1}{\sqrt{14}}\right|=\frac{1}{\sqrt{14}}$ unit
Hence, the given lines do not intersect.
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