Question

Find the shortest distance between $\overline{r}=\hat{i}+\lambda (\hat{j}+\hat{k})and\overline{r}=2\hat{j}+\beta \left(2\hat{k}\right)$.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is A 1

As we know shortest distance between $\overline{r}=\overline{a}+\lambda \overline{p}and\overline{r}=\overline{b}+\beta \overline{q}is\left|\frac{(\overline{a}-\overline{b}).(\overline{p}\times \overline{q})}{(|\overline{p}\times \overline{q}|)}\right|$ Lets substitute.
Here, for $\overline{a}and\overline{b}$ we have $\hat{i}and2\hat{j}.$
$\overline{p}=\hat{j}+\hat{k},\overline{q}=2\hat{k}\Rightarrow (\overline{a}-\overline{b}).(\overline{p}\times \overline{q})=(\hat{i}-2\hat{j}).(\overline{p}\times \overline{q})\overline{p}\times \overline{q}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 0& 1& 1\\ 0& 0& 2\end{array}\right|=\hat{i}\left(2\right)\Rightarrow (\hat{i}-2\hat{i}).\left(2\hat{i}\right)=2\Rightarrow \left|\frac{2}{\overline{p}\times \overline{q}}\right|=\left|\frac{2}{2}\right|=1units.$