Let the line of the shortest distance between the lines gathered L_1: r=(i+2j+3k)+ (i-j+k)…

Question

Let the line of the shortest distance between the lines

$\begin{gathered} {L}_{1}:\overset{	o }{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k}) \ {L}_{2}:\overset{	o }{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu (\hat{i}+\hat{j}-\hat{k}) \end{gathered}$

intersect ${L}_{1}$ and ${L}_{2}$ at $P$ and $Q$ respectively. If $(\alpha ,\beta ,\gamma )$ is the mid point of the line segment $PQ$, then $2(\alpha +\beta +\gamma )$ is equal to \_\_\_\_\_\_\_\_\_\_\_\_.

Answer: 21

Accepted Answer

$\begin{gathered} {L}_{1}\equiv \vec{r}=(1,2,3)+\lambda (1,-1,1) (\vec{r}={\vec{a}}_{1}+\lambda {\vec{b}}_{1}) \ \ {L}_{2}\equiv \vec{r}=(4,5,6)+\mu (1,1,-1) (\vec{r}={\vec{a}}_{2}+\lambda {\vec{b}}_{2}) $

$P\equiv (\lambda +1,-\lambda +2,\lambda +3)$

$Q\equiv (\mu +4,\mu +5,-\mu +6)$

$\overset{	o \leftarrow }{PQ}=(\mu -\lambda +3,\mu +\lambda +3,-\mu -\lambda +3)$

$\overset{	o \leftarrow }{PQ}\cdot {\vec{b}}_{1}=0\Rightarrow 3\lambda +\mu =3$ ..........(i)

$\overset{	o \leftarrow }{PQ}\cdot {\vec{b}}_{2}=0\Rightarrow 3\mu +\lambda =3$ ..........(ii)

From (i) and (ii),

$ P\equiv (\frac{5}{2},\frac{1}{2},\frac{9}{2})Q\equiv (\frac{5}{2},\frac{7}{2},\frac{15}{2}) \ \ \alpha =\frac{5}{2},\beta =\frac{4}{2},\gamma =\frac{12}{2} \ \ 2(\alpha +\beta +\gamma )=21 \end{gathered}$

Solutions · 1