The distance between parallel lines vec{r}=(2 | vedclass

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

Question

(D) The distance $d$ between two parallel lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}$ is given by $d = \frac{|

Vedclass · Accepted Answer

$\frac{\sqrt{2}}{3}$ units

Vedclass · Answer

(D) The distance $d$ between two parallel lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}$ is given by $d = \frac{|(\vec{a}_2 - \vec{a}_1) 	imes \vec{b}|}{|\vec{b}|}$.Here,$\vec{a}_1 = 2\hat{i} - \hat{j} + \hat{k}$,$\vec{a}_2 = \hat{i} - \hat{j} + 2\hat{k}$,and $\vec{b} = 2\hat{i} + \hat{j} - 2\hat{k}$.First,calculate $\vec{a}_2 - \vec{a}_1 = (1-2)\hat{i} + (-1 - (-1))\hat{j} + (2-1)\hat{k} = -\hat{i} + \hat{k}$.Next,calculate the cross product $(\vec{a}_2 - \vec{a}_1) 	imes \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ -1 & 0 & 1 \ 2 & 1 & -2 \end{vmatrix} = \hat{i}(0-1) - \hat{j}(2-2) + \hat{k}(-1-0) = -\hat{i} - \hat{k}$.The magnitude is $|(\vec{a}_2 - \vec{a}_1) 	imes \vec{b}| = \sqrt{(-1)^2 + 0^2 + (-1)^2} = \sqrt{2}$.The magnitude of $\vec{b}$ is $|\vec{b}| = \sqrt{2^2 + 1^2 + (-2)^2} = \sqrt{4+1+4} = \sqrt{9} = 3$.Therefore,$d = \frac{\sqrt{2}}{3}$ units.

The distance between parallel lines $\vec{r}=(2\hat{i}-\hat{j}+\hat{k})+\lambda(2\hat{i}+\hat{j}-2\hat{k})$ and $\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(2\hat{i}+\hat{j}-2\hat{k})$ is

Similar Questions

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

The distance of the point having position vector $-\hat{i} + 2\hat{j} + 6\hat{k}$ from the straight line passing through the point $(2, 3, -4)$ and parallel to the vector $6\hat{i} + 3\hat{j} - 4\hat{k}$ is

If $A(3,4,5)$,$B(4,6,3)$,$C(-1,2,4)$,and $D(1,0,5)$ are points such that the angle between the lines $DC$ and $AB$ is $\theta$,then $\cos \theta$ is equal to

The angle between the lines $3x = 2y = -z$ and $-x = 6y = -4z$ is

If the direction ratios of a line are in the ratio $2 : 1 : 2$ and it intersects the lines $x = y + a = z$ and $x + a = 2y = 2z$,then the coordinates of the points of intersection are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module