MEDIUM

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If the length of the projection of the line segment joining the points $(1,2, - 1)$ and $(3,5, 5)$ on the plane $3 x - 4 y + 12 z = 5$ is equal to $d$ units, then the value of $169 d^{2}$ is equal to

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Important Questions on Three Dimensional Geometry

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

Let

P

be the image of the point

(3, 1, 7)

with respect to the plane

x - y + z = 3 .

Then the equation of the plane passing through

P

and containing the straight line

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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

On which of the following lines lies the point of intersection of the line,

\frac{x - 4}{2} = \frac{y - 5}{2} = \frac{z - 3}{1}

and the plane,

x + y + z = 2 ?

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

If the line,

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

lies in the plane

l x + m y - z = 9,

then

l^{2} + m^{2}

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

Acute angle between the line

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

and the plane

3 x - 4 y - z + 5 = 0

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

If the angle between the line

2 (x + 1) = y = z + 4

and the plane

2 x - y + \sqrt{λ} z + 4 = 0

\frac{π}{6}

, then the value of

λ

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

If the line

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

intersects the plane

2 x + 3 y - z + 13 = 0

at a point

P

and the plane

3 x + y + 4 z = 16

at a point

Q,

then

P Q

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

Let

A

be a point on the line

\vec{r} = (1 - 3 μ) \hat{i} + (μ - 1) \hat{j} + (2 + 5 μ) \hat{k}

and

B (3, 2, 6)

be a point in the space. Then the value of

μ

for which the vector

\vec{A B}

is parallel to the plane

x - 4 y + 3 z = 1

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

If the line joining the points

(- 1, 2, 5)

and

(3, 4, - 10)

intersects the

x y

-plane at the point,

(x, y, z),

then

\frac{y}{x}

is equal to______.

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

If an angle between the line,

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

and the plane,

x - 2 y - k z = 3

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

, then a value of

k

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

The length of the projection of the line segment joining the points

(5, - 1, 4)

and

(4, - 1, 3)

on the plane,

x + y + z = 7

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

The point at which the line

\frac{x - 2}{1} = \frac{y - 4}{- 5} = \frac{z + 3}{4}

intersects the

xy

-plane is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

A plane containing the point

(3, 2, 0)

and the line

\frac{x - 1}{1} = \frac{y - 2}{5} = \frac{z - 3}{4}

also contains the point

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

The distance of the point

(1, - 2, 3)

from the plane

x - y + z = 5

measured parallel to the line

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

is :

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

If the line,

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

meets the plane,

x + 2 y + 3 z = 15

at a point

P,

then the distance of

P

from the origin is,

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

θ

is the angle between the line

\vec{r} = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (\hat{i} - \hat{j} + 2 \hat{k}), λ \in R

and the plane

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

then

a

value of

\cos θ

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

A line

L

is passing through the point

A

whose position vector is

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

and parallel to the vector

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

. A plane

π

is passing through the points

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

and parallel to the vector

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

. Then the point where this plane

π

meets the line

L

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

Line

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

is perpendicular to

2 x + 2 y - 8 z + 5 = 0

, then

λ

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

The distance of the point

(1, - 5, 9)

from the plane

x - y + z = 5

measured along the line

x = y = z

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

If the image of the point

P (1, - 2, 3)

in the plane,

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

measured parallel to the line,

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

Q

, then

P Q

is equal to:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between a Line and a Plane

The distance of the point

(1, 0, 2)

from the point of intersection of the line

\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}

and the plane

x - y + z =16

, is

If the length of the projection of the line segment joining the points 1,2,-1 and 3,5,5 on the plane 3x-4y+12z=5 is equal to d units, then the value of 169 d2 is equal to

Important Questions on Three Dimensional Geometry

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If the length of the projection of the line segment joining the points $(1,2, - 1)$ and $(3,5, 5)$ on the plane $3 x - 4 y + 12 z = 5$ is equal to $d$ units, then the value of $169 d^{2}$ is equal to