If the shortest distance between the lines array|c|c| L_1:r=(2+ )i+(1-3 )j+(3+4 )k, & R L_…

Question

If the shortest distance between the lines

$\begin{array}{|c|c|} {L}_{1}:\vec{r}=(2+\lambda )\hat{i}+(1-3\lambda )\hat{j}+(3+4\lambda )\hat{k}, & \lambda \in R \ {L}_{2}:\vec{r}=2(1+\mu )\hat{i}+3(1+\mu )\hat{j}+(5+\mu )\hat{k}, & \mu \in R \end{array}$

is $\frac{m}{\sqrt{n}}$, where $gcd(m,n)=1$, then the value of $m+n$ equals

Accepted Answer

$\begin{gathered} {L}_{1}:\vec{r}=(2+\lambda )\hat{i}+(1-3\lambda )\hat{j}+(3+4\lambda )\hat{k} \ {L}_{1}=2\hat{i}+\hat{j}+3\hat{k}+\lambda (\hat{i}-3\hat{j}+4\hat{k}) \ {L}_{2}:\vec{r}=2\hat{i}+3\hat{j}+5\hat{k}+\mu (2\hat{i}+3\hat{j}+\hat{k}) \ {\vec{a}}_{1}=2\hat{i}+\hat{j}+3\hat{k} \ {\vec{a}}_{2}=2\hat{i}+3\hat{j}+5\hat{k} \ {\vec{a}}_{2}-{\vec{a}}_{1}=2\hat{j}+2\hat{k} \ {\vec{b}}_{1}=\hat{i}-3\hat{j}+4\hat{k},{\vec{b}}_{2}=2\hat{i}+3\hat{j}+\hat{k} \ {\vec{b}}_{1}	imes {\vec{b}}_{2}=|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ 1 \quad  -3 \quad  4 \ 2 \quad  3 \quad  1\end{array}| \end{gathered}$

$\begin{gathered} \hat{i}(-3-12)-\hat{j}(1-8)+\hat{k}(3+6) \ =-15\hat{i}+7\hat{j}+9\hat{k} \ |{\vec{b}}_{1}	imes {\vec{b}}_{2}|=\sqrt{225+49+81} \ |\frac{({\vec{a}}_{2}-{\vec{a}}_{1})\cdot ({\vec{b}}_{1}	imes {\vec{b}}_{2})}{|{\vec{b}}_{1}	imes {\vec{b}}_{2}|}|=\frac{14+18}{\sqrt{355}}=\frac{32}{\sqrt{355}} \ m+n=387 \end{gathered}$

Solutions · 1