Answer:
Given equations of lines are \[\vec{r}=(3\hat{i}+5\hat{j}+7\hat{k})+\lambda (\hat{i}-2\hat{j}+\hat{k})\] and \[\vec{r}=(-\,\hat{i}-\hat{j}-\hat{k})+\mu (7\hat{i}-6\hat{j}+\hat{k})\] On comparing with \[\vec{r}={{\vec{a}}_{1}}+\lambda \,{{\vec{b}}_{1}}\] and \[\vec{r}={{\vec{a}}_{2}}+\mu {{\vec{b}}_{2}},\] we get \[{{\vec{a}}_{1}}=3\hat{i}+5\hat{j}+7\hat{k},\] \[{{\vec{b}}_{1}}=\hat{i}-2\hat{j}+\hat{k}\] \[{{\vec{a}}_{2}}=-\,\hat{i}-\hat{j}-\hat{k},\] \[{{\vec{b}}_{2}}=7\hat{i}-6\hat{j}+\hat{k}\] Now, \[{{\vec{a}}_{2}}-{{\vec{a}}_{1}}=(-\,\hat{i}-\hat{j}-\hat{k})-(3\hat{i}\,+5\hat{j}\,+7\hat{k})\] \[=-\,\hat{i}-\hat{j}-\hat{k}-3\hat{i}\,-5\hat{j}\,-7\hat{k}=-\,4\hat{i}-6\hat{j}-8\hat{k}\] and \[{{\vec{b}}_{1}}\times {{\vec{b}}_{2}}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & -\,2 & 1 \\ 7 & -\,6 & 1 \\ \end{matrix} \right|\] \[=\hat{i}(-\,2+6)-\hat{j}(1-7)+\hat{k}(-\,6+14)\] \[=4\hat{i}\,+6\hat{j}\,+8\hat{k}\] \[\therefore \] \[|{{\vec{b}}_{1}}\times {{\vec{b}}_{2}}|=\sqrt{{{(4)}^{2}}+{{(6)}^{2}}+{{(8)}^{2}}}\] \[=\sqrt{16+36+64}=\sqrt{116}\] Shortest distance between two lines is \[SD=\left| \frac{({{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}})\cdot ({{{\vec{a}}}_{2}}-{{{\vec{a}}}_{1}})}{|{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|} \right|\] \[=\left| \frac{(4\hat{i}\,+6\hat{j}\,+8\hat{k})\cdot (-\,4\hat{i}\,-6\hat{j}\,-8\hat{k})}{\sqrt{116}} \right|\] \[=\left| \frac{-\,16-36-64}{\sqrt{116}} \right|=\left| \frac{-\,116}{\sqrt{116}} \right|\] \[=\left| \frac{-\,116}{\sqrt{116}} \right|=\sqrt{116}=2\sqrt{29}\] units Since, shortest distance between them is not zero, so they will not meet to an accident. If a driver follows speed limit, then there will be minimum chance to meet with an accident.
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