Question

The distance between the parallel lines given by the equations, \[\overrightarrow{r}.(2\widehat{i}-2\widehat{j}+\widehat{k})+3=0\] and \[\overrightarrow{r}.(4\widehat{i}-4\widehat{j}+2\widehat{k})+5=0\] is
A. \[\dfrac{1}{2}\]
B. \[\dfrac{1}{6}\]
C. \[\dfrac{\sqrt{2}}{3}\]
D. \[1\]

Answer 1

Hint: the distance between two parallel lines \[\overrightarrow{r}.(a\widehat{i}+b\widehat{j}+c\widehat{k})+{{c}_{1}}=0\] and \[\overrightarrow{r}.(a\widehat{i}+b\widehat{j}+c\widehat{k})+{{c}_{2}}=0\] is given by \[\left| \dfrac{{{c}_{1}}-{{c}_{2}}}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right|\]. multiply the line equation (1) with 2 to get both as parallel as parallel lines and then apply the distance between two parallel lines formulas.

Complete step-by-step solution -
Given the line equations are \[\overrightarrow{r}.(2\widehat{i}-2\widehat{j}+\widehat{k})+3=0\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1)
                                                    \[\overrightarrow{r}.(4\widehat{i}-4\widehat{j}+2\widehat{k})+5=0\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .(2)
Multiply the equation (1) with 2 then we will get
\[\overrightarrow{r}.(4\widehat{i}-4\widehat{j}+2\widehat{k})=-6\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
Rewriting equation (2) as same
\[\overrightarrow{r}.(4\widehat{i}-4\widehat{j}+2\widehat{k})=-5\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
We know that the distance between two parallel lines is given by \[\left| \dfrac{{{c}_{1}}-{{c}_{2}}}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right|\]
\[=\left| \dfrac{-6-(-5)}{\sqrt{{{4}^{2}}+{{4}^{2}}+{{2}^{2}}}} \right|\]
\[=\left| \dfrac{-1}{\sqrt{16+16+4}} \right|\]
\[=\dfrac{1}{6}\]
So, the distance between the parallel lines given by equations \[\overrightarrow{r}.(2\widehat{i}-2\widehat{j}+\widehat{k})+3=0\] and \[\overrightarrow{r}.(4\widehat{i}-4\widehat{j}+2\widehat{k})+5=0\] is \[\dfrac{1}{6}\]
So, the correct option is option (B)

Note: In geometry, the parallel lines are lines which do not intersect at any point in a plane. Skew lines and parallel lines are different; note that skew lines never meet and they are not parallel. The parallel lines equation differ only in constant value and values of a, b, c that is the coefficient of x, y, z of two parallel lines are same