The lines bar{r} = (hat{i} + hat{j} - hat{k}) | vedclass

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

Question

(A) The given lines are in the form $\bar{r} = \bar{a}_1 + \lambda \bar{b}_1$ and $\bar{r} = \bar{a}_2 + \mu \bar{b}_2$. Here,$\bar{b}_1 = 3 \hat{i} -

Vedclass · Accepted Answer

intersecting but not perpendicular

Vedclass · Answer

(A) The given lines are in the form $\bar{r} = \bar{a}_1 + \lambda \bar{b}_1$ and $\bar{r} = \bar{a}_2 + \mu \bar{b}_2$. Here,$\bar{b}_1 = 3 \hat{i} - \hat{j}$ and $\bar{b}_2 = 2 \hat{i} + 3 \hat{k}$. First,check if the lines are parallel: $\bar{b}_1$ is not a scalar multiple of $\bar{b}_2$,so they are not parallel. Next,check for intersection by equating the lines: $(\hat{i} + \hat{j} - \hat{k}) + \lambda(3 \hat{i} - \hat{j}) = (4 \hat{i} - \hat{k}) + \mu(2 \hat{i} + 3 \hat{k})$. Equating components: $1 + 3\lambda = 4 + 2\mu \implies 3\lambda - 2\mu = 3$ $1 - \lambda = 0 \implies \lambda = 1$ $-1 = -1 + 3\mu \implies 3\mu = 0 \implies \mu = 0$ Substituting $\lambda = 1$ and $\mu = 0$ into the first equation: $3(1) - 2(0) = 3$,which is $3 = 3$. Since the equations are consistent,the lines intersect. Check for perpendicularity: $\bar{b}_1 \cdot \bar{b}_2 = (3)(2) + (-1)(0) + (0)(3) = 6 
eq 0$. Thus,the lines are intersecting but not perpendicular.

The lines $\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3 \hat{i} - \hat{j})$ and $\bar{r} = (4 \hat{i} - \hat{k}) + \mu(2 \hat{i} + 3 \hat{k})$ are

Similar Questions

The shortest distance between the lines $\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$ and $x+y+z+1=0, 2x-y+z+3=0$ is

$A$ line $L$ passes through points $A(1, 3, 2)$ and $B(2, 2, 1)$. If the mirror image of point $P(1, 1, -1)$ in the line $L$ is $(x, y, z)$,then $x+y+z=$

The distance of the point $P(7, 10, 11)$ from the line $\frac{x-4}{1} = \frac{y-4}{0} = \frac{z-2}{3}$ along the line $\frac{x-9}{2} = \frac{y-13}{3} = \frac{z-17}{6}$ is

The angle between the lines $\frac{x}{1} = \frac{y}{0} = \frac{z}{-1}$ and $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$ is

The equation of the line passing through the point $(1, -3, 5)$ and making equal angles with the coordinate axes is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module