Let l 0 be the line defined by the vector (equation) i ^ + 2 j ^ + 3 k ^ + λ ( i ^ + j ^ + k ^ ) , with λ real. Which of the following vector equations, with μ real, defines a line which intersects l 0 ?

i ^ + 3 j ^ + 5 k ^ + μ ( 2 i ^ + 3 j ^ + 4 k ^ )

- i ^ - 2 j ^ - 3 k ^ + μ ( - i ^ - j ^ - k ^ )

3 i ^ - 2 j ^ + k ^ + μ ( - i ^ + j ^ )

2 i ^ + 3 j ^ + μ ( i ^ - j ^ )

Let l 0 be the line defined by the vector (equation) i ^ + 2 j ^ + 3 k ^ + λ ( i ^ + j ^ + k ^ ) , with λ real. Which of the following vector equations, with μ real, defines a line which intersects l 0 ?

i ^ + 3 j ^ + 5 k ^ + μ ( 2 i ^ + 3 j ^ + 4 k ^ )

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

The lines r → = i ^ - j ^ + l 2 i ^ + k ^ and r → = 2 i ^ - j ^ + m i ^ + j ^ - k ^

Do not intersect for any values of l and m

Intersect for all values of l and m

Intersect when l = 2 and m = 1 2

EASY

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Let $l_{0}$ be the line defined by the vector (equation) $\hat{i} + 2 \hat{j} + 3 \hat{k} + λ (\hat{i} + \hat{j} + \hat{k})$ , with $λ$ real. Which of the following vector equations, with $μ$ real, defines a line which intersects $l_{0}$ ?

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Important Questions on Three Dimensional Geometry

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

The angle between the lines whose direction cosines satisfy the equations

l + m + n = 0

and

l^{2} = m^{2} + n^{2}

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

The number of distinct real values of

λ

, for which the lines

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

and

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

, are coplanar is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

If line

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

and

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

intersect each other, then

λ = . \dots .

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

Let

A (3, 0, - 1), B (2, 10, 6)

and

C (1, 2, 1)

be the vertices of a triangle and

M

be the mid-point of

A C

. If

G

divides

B M

in the ratio,

2 : 1

, then

c o s (∠ G O A)

(

O

being the origin) is equal to

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

If for some,

α \in R,

the lines

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

and

L_{2} : \frac{x + 2}{α} = \frac{y + 1}{5 - α} = \frac{z + 1}{1}

are coplanar, then the line

L_{2}

passes through the point :

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

If the angle between the lines

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

and

\frac{5 - x}{- 2} = \frac{7 y - 14}{P} = \frac{z - 3}{4}

\cos^{- 1} (\frac{2}{3}),

then

P

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

If lines

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

and

\frac{x - 1}{3 k} = \frac{y - 5}{1} = \frac{z - 6}{- 5}

are mutually perpendicular, then

k

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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}

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

If the lines

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

and

\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}

are coplanar, then

k

can have

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

If the lines

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

and

\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}

are coplanar, then

k

can have

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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 ____________.

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

The angle between the line

A B

and

C D

, where

A = (0, 0, 0), B = (1, 1, 1), C = (- 1, - 1, - 1)

and

D = (0, 1, 0)

is given by

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

The lines

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

and

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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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}

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between 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 :

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Lines in 3D

If the lines

x = a y + b, z = c y + d

and

x = a^{'} z + b^{'}, y = c^{'} z + d^{'}

are perpendicular, then

Let l0 be the line defined by the vector (equation) i^+2j^+3k^+λ(i^+j^+k^), with λ real. Which of the following vector equations, with μ real, defines a line which intersects l0 ?

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

Let $l_{0}$ be the line defined by the vector (equation) $\hat{i} + 2 \hat{j} + 3 \hat{k} + λ (\hat{i} + \hat{j} + \hat{k})$ , with $λ$ real. Which of the following vector equations, with $μ$ real, defines a line which intersects $l_{0}$ ?