Equation of the line of the shortest distance between the lines x ⁡ 1 = y ⁡ - 1 = z ⁡ 1 and x ⁡ - 1 0 = y ⁡ + 1 - 2 = z ⁡ 1 is

x ⁡ - 1 1 = y ⁡ + 1 - 1 = z ⁡ - 2

x ⁡ 1 = y ⁡ - 1 = z ⁡ - 2

x ⁡ - 1 1 = y ⁡ + 1 - 1 = z ⁡ 1

HARD

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The shortest distance between lines $L_{1}$ and $L_{2}$ , where $L_{1} : \frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 4}{2}$ and $L_{2}$ is the line passing through the points $A (- 4, 4, 3), B (- 1, 6, 3)$ and perpendicular to the line $\frac{x - 3}{- 2} = \frac{y}{3} = \frac{z - 1}{1}$ , is

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Important Questions on Three Dimensional Geometry

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the lines

\vec{r} = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 3 \hat{k})

and

\vec{r} = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (2 \hat{i} + 4 \hat{j} - 5 \hat{k})

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

If the shortest distance between the line

\frac{x - 1}{α} = \frac{y + 1}{- 1} = \frac{z}{1}, (α \neq - 1)

, and

x + y + z + 1 = 0 = 2 x - y + z + 3

\frac{1}{\sqrt{3}}

,then value of

α

is :

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the lines

\vec{r} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2 t) \hat{k}

and

\vec{r} = (p + 1) \hat{i} + (2 p - 1) \hat{ȷ} + (2 p + 1) \hat{k}

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The value of

α

for which the shortest distance between the lines represented by

y + z = 0, z + x = 0

and

x + y = 0, x + y + z = α

1,

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

If the distance between the plane,

23 x - 10 y - 2 z + 48 = 0

and the plane containing the lines

\frac{x + 1}{2} = \frac{y - 3}{4} = \frac{z + 1}{3}

and

\frac{x + 3}{2} = \frac{y + 2}{6} = \frac{z - 1}{λ} (λ \in R)

is equal to

\frac{k}{\sqrt{633}},

then

k

is equal to ____________.

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

If the shortest distance between the lines

\vec{r_{1}} = α \hat{i} + 2 \hat{j} + 2 \hat{k} + λ (\hat{i} - 2 \hat{j} + 2 \hat{k}), λ \in R, α > 0

and

\vec{r_{2}} = - 4 \hat{i} - \hat{k} + μ (3 \hat{i} - 2 \hat{j} - 2 \hat{k}), μ \in R

9,

then

α

is equal to_____.

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

\vec{r} = \hat{i} + \hat{j} + t (2 \hat{i} - \hat{j} + \hat{k})

and

\vec{r} = 2 \hat{i} - \hat{j} - \hat{k} + t (3 \hat{i} - 5 \hat{j} + 2 \hat{k})

are the vector equations of two lines

L_{1}

and

L_{2}

then the shortest distance between them is

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the lines

\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 6}{2}

and

\frac{x - 6}{3} = \frac{1 - y}{2} = \frac{z + 8}{0}

is equal to ______

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the

z

- axis and the line

x + y + 2 z - 3 = 0 = 2 x + 3 y + 4 z - 4,

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the lines

\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 3}{1}

and

\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

If the shortest distance between the lines

\vec{r} = (- \hat{i} + 3 \hat{k}) + λ (\hat{i} - a \hat{j})

and

\vec{r} = (- \hat{j} + 2 \hat{k}) + μ (\hat{i} - \hat{j} + \hat{k})

\sqrt{\frac{2}{3}}

, then the integral value of

a

is equal to _____

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

Equation of the line of the shortest distance between the lines

\frac{x}{1} = \frac{y}{- 1} = \frac{z}{1}

and

\frac{x - 1}{0} = \frac{y + 1}{- 2} = \frac{z}{1}

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

If the shortest distance between the line joining the points

(1, 2, 3)

and

(2, 3, 4)

, and the line

\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 2}{0}

α

, then

28 α^{2}

is equal to _____ .

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

Let

λ

be an integer. If the shortest distance between the lines

x - λ = 2 y - 1 = - 2 z

and

x = y + 2 λ = z - λ

\frac{\sqrt{7}}{2 \sqrt{2}},

then the value of

|λ|

is _______.

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

If the shortest distance between the lines

\frac{x + \sqrt{6}}{2} = \frac{y - \sqrt{6}}{3} = \frac{z - \sqrt{6}}{4}

and

\frac{x - λ}{3} = \frac{y - 2 \sqrt{6}}{4} = \frac{z + 2 \sqrt{6}}{5}

6

, then square of sum of all possible values(s) of

λ

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The line

l_{1}

passes through the point

(2, 6, 2)

and is perpendicular to the plane

2 x + y - 2 z = 10

. Then the shortest distance between the line

l_{1}

and the line

\frac{x + 1}{2} = \frac{y + 4}{- 3} = \frac{z}{2}

is:

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the lines

x + 1 = 2 y = - 12 z

and

x = y + 2 = 6 z - 6

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the lines

\frac{x - 1}{0} = \frac{y + 1}{- 1} = \frac{z}{1}

and

x + y + z + 1 = 0, 2 x - y + z + 3 = 0

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

The shortest distance between the lines

\frac{x}{2} = \frac{y}{2} = \frac{z}{1}

and

\frac{x + 2}{- 1} = \frac{y - 4}{8} = \frac{z - 5}{4},

lies in the interval:

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of two Lines in 3D

If the line,

\frac{x - 3}{1} = \frac{y + 2}{- 1} = \frac{z + λ}{- 2}

lies in the plane,

2 x - 4 y + 3 z = 2

, then the shortest distance between this line and the line,

\frac{x - 1}{12} = \frac{y}{9} = \frac{z}{4}

The shortest distance between lines L1 and L2, where L1:x−12=y+1−3=z+42 and L2 is the line passing through the points A−4,4,3, B−1,6,3 and perpendicular to the line x−3−2=y3=z−11, is

Important Questions on Three Dimensional Geometry

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The shortest distance between lines $L_{1}$ and $L_{2}$ , where $L_{1} : \frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 4}{2}$ and $L_{2}$ is the line passing through the points $A (- 4, 4, 3), B (- 1, 6, 3)$ and perpendicular to the line $\frac{x - 3}{- 2} = \frac{y}{3} = \frac{z - 1}{1}$ , is