HARD

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Let $L_{1} : \vec{r} = (\hat{i} + \hat{j} + \hat{k}) + λ (\hat{i} + \hat{j}), λ \in R$ $L_{2} : \hat{i} + (μ + 1) \hat{j} + (μ + 1) \hat{k}, μ \in R$ are two lines intersecting at point $' A' .$ Through point $B (3, 3, 1)$ a line in drawn making an angle of $60 °$ with $L_{2}$ and intersecting it at point $' C' .$ Then area of $Δ A B C$ is

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Important Questions on Three Dimensional Geometry

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

The vector equation of the line passing through the point

(- 1, 5, 4)

and perpendicular to the plane

z = 0

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

The Cartesian equation of the line passing through the points

(1, - 1, 2)

and

(7, 0, 5)

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

The equation of the line joining the points

((- 3, 4, 11)

and

(1, - 2, 7)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

The Cartesian equation of line passing through the points

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

is_____

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

The parametric equations of the line passing through

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

are

(λ \in R)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line in 3D

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>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of a Line 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 .

Let L1:r→=i^+j^+k^+λi^+j^, λ∈R L2:i^+μ+1j^+μ+1k^, μ∈R are two lines intersecting at point 'A'. Through point B3,3,1 a line in drawn making an angle of 60° with L2 and intersecting it at point 'C'. Then area of ΔABC is

Important Questions on Three Dimensional Geometry

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