Question

Find the distance between the two given lines.

$$\begin{gathered} \frac{\mathbf{x}\mathbf{-}\mathbf{1}}{\mathbf{2}}\mathbf{=}\frac{\mathbf{y}\mathbf{+}\mathbf{2}}{\mathbf{3}}\mathbf{=}\frac{\mathbf{2}\mathbf{z}\mathbf{-}\mathbf{1}}{\mathbf{4}}\mathbf{and} \\ \frac{\mathbf{x}\mathbf{+}\mathbf{2}}{\mathbf{-}\mathbf{1}}\mathbf{=}\frac{\mathbf{2}\mathbf{-}\mathbf{y}}{\mathbf{2}}\mathbf{,}\mathbf{z}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}} \end{gathered}$$

Short answer

The distance between both the lines are$\frac{20}{\sqrt{21}}$.

Step-by-step solution

Given equation

The given lines are$\frac{x-1}{2}=\frac{y+2}{3}=\frac{2z-1}{4}\mathrm{and}\frac{\mathrm{x}+2}{-1}=\frac{2-\mathrm{y}}{2},\mathrm{z}=\frac{1}{2}.$.

Concept used

Introduction of a unit vector perpendicular to both the lines and then solving for the value of the distance between the given vector and the unit vector gives the distance between the given lines.

To calculate the distance d between two skew lines

Then $\left|PQ.n\right|$where n is a unit vector perpendicular to both lines, is the distance d . Now if A and A are vectors along the two lines, then $\mathrm{A}\times \mathrm{B}$is perpendicular to both, and n is just $\mathrm{A}\times \mathrm{B}$divided by $\left|\mathrm{A}\times \mathrm{B}\right|$. In this problem, consider the two points as following:

The symmetric form role="math" localid="1658815111418" $\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$.

So, deduce that $\mathrm{P}=\left(1,-2\frac{1}{2}\right)$and $Q=\left(-2,2,\frac{1}{2}\right)$which mean

$PQ=\left(1,-2,\frac{1}{2}\right)-\left(-2,2,\frac{1}{2}\right)=\left(-3,4,0,\right)$

Given that a vector V=ai+bj+ck is parallel to the line described by the equation

$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$

The two vectors along the lines are as follows:

A=2i+3j+2k

B=-i-2j

$$\begin{gathered} \mathrm{A}\times \mathrm{B}=\left(\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 3& 2\\ -1& -2& 0\end{array}\right) \\ =4\mathrm{i}-2\mathrm{j}-\mathrm{k} \\ \mathrm{n}=\frac{\mathrm{A}\times \mathrm{B}}{\left|\mathrm{A}\times \mathrm{B}\right|} \\ =\frac{4\mathrm{i}-2\mathrm{j}-\mathrm{k}}{\sqrt{{\left(4\right)}^2+{\left(-2\right)}^2+{\left(-1\right)}^2}} \\ =\frac{4\mathrm{i}-2\mathrm{j}-\mathrm{k}}{\sqrt{21}} \end{gathered}$$

Calculation of distance

The formula for distance between two lines is given as follows:

$\mathrm{d}=\left|\mathrm{PQ}.\mathrm{n}=\frac{1}{\sqrt{21}}\left|\left[\left(-3\right)\left(4\right)+\left(4\right)\left(-2\right)+\left(0\right)\left(-1\right)\right]\right|=\frac{1}{\sqrt{21}}-20\right|=\frac{20}{\sqrt{21}}$

The distance between the two lines is $\frac{20}{\sqrt{21}}$.