(B) The direction ratios $(l, m, n)$ of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are given by $(x_2 - x_1, y_2 - y_1,

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(B) The direction ratios $(l, m, n)$ of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are given by $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$.Given points are $(1, 2, -1)$ and $(-1, 0, 1)$.Calculating the differences:$l = -1 - 1 = -2$$m = 0 - 2 = -2$$n = 1 - (-1) = 2$Thus,the direction ratios are proportional to $(-2, -2, 2)$.Dividing by $-2$,we get the ratios $(1, 1, -1)$.Therefore,$(l, m, n) = (1, 1, -1)$.

Class 12 Mathematics THREE DIMENSIONAL GEOMETRY Line

Easy

If $\frac{x - 1}{l} = \frac{y - 2}{m} = \frac{z + 1}{n}$ is the equation of the line passing through the points $(1, 2, -1)$ and $(-1, 0, 1)$,then the direction ratios $(l, m, n)$ are:

A
$(-1, 0, 1)$
B
$(1, 1, -1)$
C
$(1, 2, -1)$
D
$(0, 1, 0)$

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THREE DIMENSIONAL GEOMETRY

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Line

If the $x$-coordinate of a point $P$ on the line joining the points $Q(2, 2, 1)$ and $R(5, 2, -2)$ is $4$,then the $y$-coordinate of $P$ is:

EasyAP EAMCET 2022

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Let the line $L$ pass through the point $(-3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of $L$ from the point $P(-2, r, 1)$ is $\sqrt{\frac{14}{3}}$, then the sum of all possible values of $r$ is:

DifficultJEE MAIN 2026

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

Medium

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Let the shortest distance between the lines $\frac{x-3}{3}=\frac{y-\alpha}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta}{4}$ be $3 \sqrt{30}$. Then the positive value of $5 \alpha+\beta$ is

DifficultJEE MAIN 2025

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The shortest distance between the lines $\overline{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\overline{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(2\hat{i} + 4\hat{j} - 5\hat{k})$ is:

MediumMHT CET 2025

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If $\frac{x - 1}{l} = \frac{y - 2}{m} = \frac{z + 1}{n}$ is the equation of the line passing through the points $(1, 2, -1)$ and $(-1, 0, 1)$,then the direction ratios $(l, m, n)$ are:

Similar Questions

If the $x$-coordinate of a point $P$ on the line joining the points $Q(2, 2, 1)$ and $R(5, 2, -2)$ is $4$,then the $y$-coordinate of $P$ is:

Let the line $L$ pass through the point $(-3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of $L$ from the point $P(-2, r, 1)$ is $\sqrt{\frac{14}{3}}$, then the sum of all possible values of $r$ is:

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

Let the shortest distance between the lines $\frac{x-3}{3}=\frac{y-\alpha}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta}{4}$ be $3 \sqrt{30}$. Then the positive value of $5 \alpha+\beta$ is

The shortest distance between the lines $\overline{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\overline{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(2\hat{i} + 4\hat{j} - 5\hat{k})$ is:

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