Question

Find the shortest distance between the lines $\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$ and $\overrightarrow{r}=2\hat{i}+4\hat{j}+5\hat{k}+\mu (4\hat{i}+6\hat{j}+8\hat{k})$.

Answer 1

Let $l_1$ and $l_2$ be the given lines whose equations are
$\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b}$
and $\overrightarrow{r}=\overrightarrow{a_2}+2\mu \overrightarrow{b}$

Through the points $\overrightarrow{a_1}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{a_2}=2\hat{i}+4\hat{j}+5\hat{k}$ are parallel to $\overrightarrow{b}=2\hat{i}+3\hat{j}+4\hat{k}$.

Hence the distance between the lines using the formula :
$\frac{|\overrightarrow{b}\times (a_2-a_1)|}{|\overrightarrow{b}|}=\frac{|(2\hat{i}+3\hat{j}+4\hat{k})\times (\hat{i}+2\hat{j}+2\hat{k})|}{|\overrightarrow{b}|}$

$=\frac{|(2\hat{i}+3\hat{j}+4\hat{k})\times (\hat{i}+2\hat{j}+2\hat{k})|}{|2\hat{i}+3\hat{j}+4\hat{k}|}$

$=\frac{\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 4\\ 1& 2& 2\end{array}\right|}{\sqrt{4+9+16}}=\frac{|-2\hat{i}+\hat{k}|}{\sqrt{29}}$

Ans. $=\frac{\sqrt{5}}{\sqrt{2}9}$