Find the shortest distance between the lines | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Comparing $(1)$ and $(2)$ with $\vec{r}=\vec{a}_{1}+\lambda \vec{b}_{1}$ and $\vec{r}=\vec{a}_{2}+\mu \vec{b}_{2}$ respectively,we get $\vec{a}_{1

Vedclass · Accepted Answer

$\frac{10}{\sqrt{59}}$

Vedclass · Answer

(A) Comparing $(1)$ and $(2)$ with $\vec{r}=\vec{a}_{1}+\lambda \vec{b}_{1}$ and $\vec{r}=\vec{a}_{2}+\mu \vec{b}_{2}$ respectively,we get $\vec{a}_{1}=\hat{i}+\hat{j}, \quad \vec{b}_{1}=2 \hat{i}-\hat{j}+\hat{k}$$\vec{a}_{2}=2 \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}_{2}=3 \hat{i}-5 \hat{j}+2 \hat{k}$Therefore,$\vec{a}_{2}-\vec{a}_{1}=(2-1)\hat{i}+(1-1)\hat{j}+(-1-0)\hat{k} = \hat{i}-\hat{k}$Now,$\vec{b}_{1} 	imes \vec{b}_{2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & -1 & 1 \ 3 & -5 & 2 \end{vmatrix} = \hat{i}(-2+5) - \hat{j}(4-3) + \hat{k}(-10+3) = 3\hat{i} - \hat{j} - 7\hat{k}$Magnitude $|\vec{b}_{1} 	imes \vec{b}_{2}| = \sqrt{3^2 + (-1)^2 + (-7)^2} = \sqrt{9+1+49} = \sqrt{59}$The shortest distance $d$ is given by $d = \left| \frac{(\vec{b}_{1} 	imes \vec{b}_{2}) \cdot (\vec{a}_{2}-\vec{a}_{1})}{|\vec{b}_{1} 	imes \vec{b}_{2}|} ight|$$d = \left| \frac{(3\hat{i} - \hat{j} - 7\hat{k}) \cdot (\hat{i} - \hat{k})}{\sqrt{59}} ight| = \left| \frac{3(1) + (-1)(0) + (-7)(-1)}{\sqrt{59}} ight| = \left| \frac{3+0+7}{\sqrt{59}} ight| = \frac{10}{\sqrt{59}}$

Find the shortest distance between the lines $l_{1}$ and $l_{2}$ whose vector equations are
$\vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})$ $(1)$
and $\vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k})$ $(2)$

Similar Questions

If $A(1, 0, 2)$,$B(2, 1, 0)$,$C(2, -5, 3)$,and $D(0, 3, 2)$ are four points and the point of intersection of the lines $AB$ and $CD$ is $P(a, b, c)$,then $a + b + c =$

The point collinear with $(1, -2, -3)$ and $(2, 0, 0)$ among the following is

The ratio in which the $YZ$-plane divides the line segment joining the points $(2, 4, 5)$ and $(3, 5, -4)$ is:

The equation of a line passing through $(3, -1, 2)$ and perpendicular to the lines $\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})$ and $\bar{r} = (2\hat{i} + \hat{j} - 3\hat{k}) + \mu(\hat{i} - 2\hat{j} + 2\hat{k})$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Find the shortest distance between the lines $l_{1}$ and $l_{2}$ whose vector equations are$\vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})$ $(1)$and $\vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k})$ $(2)$