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The distance of the point $3 \hat{i} + 5 \hat{k}$ from the line parallel to vector $6 \hat{i} + \hat{j} - 2 \hat{k}$ and passing through the point $8 \hat{i} + 3 \hat{j} + \hat{k}$ is

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Important Questions on Three Dimensional Geometry

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The distance of the point

(1, 2, - 4)

from the line

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

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The foot of the perpendicular drawn from the point

(4, 2, 3)

to the line joining the points

(1, - 2, 3)

and

(1, 1, 0)

lies on the plane

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of perpendicular drawn from the point

(2, 3, 4)

to line

\frac{4 - x}{2} = \frac{y}{6} = \frac{1 - z}{3}

, is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The image of the point

(1, 6, 3)

in the line

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

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The distance of the point having position vector

- \hat{i} + 2 \hat{j} + 6 \hat{k}

from the straight line passing through the point

(2, 3, - 4)

and parallel to the vector,

6 \hat{i} + 3 \hat{j} - 4 \hat{k}

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of perpendicular from the point

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

to the line

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

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The image of the point

(1, 6, 3)

in the line

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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

Let the foot of perpendicular from a point

P (1, 2, - 1)

to the straight line

L : \frac{x}{1} = \frac{y}{0} = \frac{z}{- 1}

N

. Let a line be drawn from

P

parallel to the plane

x + y + 2 z = 0

which meets

L

at point

Q

. If

α

is the acute angle between the lines

P N

and

P Q

, then

\cos α

is equal to .

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

If the length of the perpendicular from the point

(β, 0, β), (β \neq 0)

to the line,

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

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

, then

β

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The distance of the point

(1, 2, 1)

from the line

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

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

Let

a, b \in R .

If the mirror image of the point

P (a, 6, 9)

with respect to the line

\frac{x - 3}{7} = \frac{y - 2}{5} = \frac{z - 1}{- 9}

(20, b, - a - 9),

then

| a + b |

is equal to:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of the perpendicular from the point

(2, - 1, 4)

on the straight line

\frac{x + 3}{10} = \frac{y - 2}{- 7} = \frac{z}{1}

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The shortest distance between

A (1, 0, 2)

and the line

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

is given by line joining

A

and

B

, then

B

in the line is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

If the position vectors of three points

A, B, C

respectively are

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

and

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

, then the perpendicular distance of the point

C

from the line

A B

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

Consider the line

L

given by the equation

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

Let

Q

be the mirror image of the point

(2, 3, - 1)

with respect to

L .

Let a plane

P

be such that it passes through

Q,

and the line

L

is perpendicular to

P .

Then which of the following points is on the plane

P ?

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

What will be the distance of point

(2, 4, - 1)

from the straight line given by

(x + 5) = (y + 3) / 4 = (6 - z) / 9 ?

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The length of the perpendicular from the point

(1, 2, 3)

to the line

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

is _________.

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

(a, b, c)

is the image of the point

(1, 2, - 3)

in the line,

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

then

a + b + c

is equal to:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

If the foot of the perpendicular drawn from the point

(1, 0, 3)

on a line passing through

(α, 7,1)

(\frac{5}{3}, \frac{7}{3}, \frac{17}{3})

, then

α

is equal to

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Line and a Point

The vertices

B

and

C

of a

Δ A B C

lie on the line,

\frac{x + 2}{3} = \frac{y - 1}{0} = \frac{z}{4}

such that

B C = 5

units. Then the area (in sq. units) of this triangle, given the point

A (1, - 1, 2),

The distance of the point 3i^+5k^ from the line parallel to vector 6i^+j^-2k^ and passing through the point 8i^+3j^+k^ is

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

The distance of the point $3 \hat{i} + 5 \hat{k}$ from the line parallel to vector $6 \hat{i} + \hat{j} - 2 \hat{k}$ and passing through the point $8 \hat{i} + 3 \hat{j} + \hat{k}$ is