The shortest distance between lines $\bar{r}=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})$ and $\bar{r}=(2 \hat{i}-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is

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Aicharya · Accepted Answer

$\begin{aligned} & \overline{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}}) 	ext { and } \ & \overline{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}) \ & 	ext { Let } \mathrm{a}_1=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \mathrm{a}_2=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}} \ & \mathrm{b}_1=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}}, \mathrm{b}_2=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}} \ & \overline{\mathrm{a}_2}-\overline{\mathrm{a}_1}=\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \ & \overline{\mathrm{b}}_1 	imes \overline{\mathrm{b}}_2=-2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} \ & \left|\overline{\mathrm{b}}_1 	imes \overline{\mathrm{b}}_2ight|=\sqrt{38}\end{aligned}$ $\begin{aligned} 	ext { Shortest distance } & =\left|\frac{\left(\overline{\mathrm{b}}_1 	imes \overline{\mathrm{b}}_2ight) \cdot\left(\overline{\mathrm{a}}_2-\overline{\mathrm{a}}_1ight)}{\left|\overline{\mathrm{b}}_1 	imes \overline{\mathrm{b}}_2ight|}ight| \ & =\left|\frac{4}{\sqrt{38}}ight|=\left|\frac{2 \sqrt{2}}{\sqrt{19}}ight| 	ext { units }\end{aligned}$

The shortest distance between lines $\bar{r}=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+\hat{j}-3…

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