निम्न दो रेखाओं के बीच न्यूनतम दूरी ज्ञात कीजिए : r = (4i - j) + λ(i + 2j - 3k) और r = (i - j + 2k) + μ(2i + 4j - 5k)

Question

निम्न दो रेखाओं के बीच न्यूनतम दूरी ज्ञात कीजिए :

\(\vec{r} =(4 \hat{i}-\hat{j})+\lambda(\hat{i}+2 \hat{j}-3 \hat{k})\) और

\(\vec{r} =(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+4 \hat{j}-5 \hat{k})\)

Answer 1

\(\vec{r}=(4 \hat{i}-\hat{j})+\lambda(\hat{i}+2 \hat{j}-5 \hat{k}) \)

\(\Rightarrow \vec{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}\)

\( \overrightarrow{a_1}=4 \hat{i}-\hat{j}, \overrightarrow{b_2}=\hat{i}+2 \hat{j}-3 \hat{k} \)

and \(\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\mu(2 \hat{i}+4 \hat{j}-5 \hat{k}) \)

\(\Rightarrow \vec{r}=\overrightarrow{a_2}+\mu \overrightarrow{b_2} \Rightarrow \vec{a}_2=\hat{i}-\hat{j}+2 \hat{k}, \overrightarrow{b_2}=2 \hat{i}+4 \hat{j}-5 \hat{k}\)

Now, \(\overrightarrow{a_2}-\overrightarrow{a_1}=\hat{i}-\hat{j}+2 \hat{k}-(4 \hat{i}-\hat{j})=3 \hat{i}+2 \hat{k}\)

\(\overrightarrow{a_1} \times \overrightarrow{a_2}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -3 \\ 2 & 4 & -5 \end{array}\right|\)