EASY

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Let $L_{1}$ be the line joining $(0, 0, 0)$ and $(1, 2, 3)$ and $L_{2}$ be the line joining $(2, 3, 4)$ and $(3, 4, 5)$ . The point of intersection of $L_{1}$ and $L_{2}$ is

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Important Questions on Line and Plane

HARD

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

Let the position vectors of two points

P

and

Q

3 \hat{i} - \hat{j} + 2 \hat{k}

and

\hat{i} + 2 \hat{j} - 4 \hat{k}

, respectively. Let

R

and

S

be two points such that the direction ratios of lines

P R

and

Q S

are

(4, - 1, 2)

and

(- 2, 1, - 2)

, respectively. Let lines

P R

and

Q S

intersect at

T

. If the vector

\vec{T A}

is perpendicular to both

\vec{P R}

and

\vec{Q S}

and the length of vector

\vec{T A}

\sqrt{5}

units, then the modulus of a position vector of

A

is :

EASY

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The equation of the line passing through the point

(2, 3, - 4)

and perpendicular to the

X O Z

plane is

(λ \in R)

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

If the lines

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

and

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

intersect, then

k

equals

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The distance of the point

(1, 2, 3)

from the plane

x + y + z = 2

measured parallel to the line

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

HARD

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The lines

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

and

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

intersect each other at point

HARD

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

For real numbers

α

and

β \neq 0,

if the point of intersection of the straight lines

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

and

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

lies on the plane

x + 2 y - z = 8,

then

α - β

is equal to :

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

A point

P

lies on a line through

Q (1, - 2, 3)

and is parallel to the line

\frac{x}{1} = \frac{y}{4} = \frac{z}{5} .

P

lies on the plane

2 x + 3 y - 4 z + 22 = 0,

then segment

P Q

equals

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The vector equation of the line passing through the point

(- 1, 5, 4)

and perpendicular to the plane

z = 0

EASY

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

Let

P

be the point of intersection of two lines

\frac{x + 10}{1} = \frac{y - 21}{7} = \frac{z + 11}{5}

and

\frac{x - 1}{5} = \frac{y - 46}{9} = \frac{z}{3} .

Q

be the point

(- 10, 21, - 11);

then

P Q

is equal to

HARD

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

Let

3 \hat{i} + \hat{j} - \hat{k}

be the position vector of a point

B

. Let

A

be a point on the line which is passing through

B

and parallel to the vector

2 \hat{i} - \hat{j} + \hat{k}

. If

|\vec{B A}| = 18

, then the position vector of

A

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

Two lines in a three-dimensional system are given as

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

and

\frac{(x - 4)}{5} = \frac{(y - 1)}{2} = z .

Their point of intersection is

EASY

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The Cartesian equation of the line passing through the points

(1, - 1, 2)

and

(7, 0, 5)

HARD

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The direction ratios of a normal to a plane are

(3, 12, 4) .

Through which of the following points the plane passes if origin is at a distance of

2

units from this plane?

EASY

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The line

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

intersects the curve

x y = c^{2}, z = 0

c

is equal to

EASY

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The equation of the line joining the points

((- 3, 4, 11)

and

(1, - 2, 7)

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

Lines

L; \frac{x + 2}{3} = \frac{y - 2}{- 1}, z + 1 = 0

and

M : {4 + 2 k, 0, - 1 + 3 k) / k \in R}

then

L \cap M

EASY

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The Cartesian equation of line passing through the points

A (4, 2, 1) and B (2, - 1, 3)

is_____

EASY

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

The parametric equations of the line passing through

A (3, 4, - 7), B (1, - 1, 6)

are

(λ \in R)

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

A plane

P

meets the coordinate axes at

A, B

and

C

respectively. The centroid of

∆ ABC

is given to be

(1, 1, 2)

. Then the equation of the line through this centroid and perpendicular to the plane

P

is :

MEDIUM

Mathematics>Coordinate Geometry>Line and Plane>Vector equation of a Line and Related Concepts

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 .

Let L1 be the line joining 0,0,0 and 1,2,3 and L2 be the line joining 2,3,4 and 3,4,5. The point of intersection of L1 and L2 is

Important Questions on Line and Plane

Learn and Prepare for any exam you want !

Let $L_{1}$ be the line joining $(0, 0, 0)$ and $(1, 2, 3)$ and $L_{2}$ be the line joining $(2, 3, 4)$ and $(3, 4, 5)$ . The point of intersection of $L_{1}$ and $L_{2}$ is