MEDIUM

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Consider the following statements:

1. Line joining $(1, 2, 5) & (4, 3, 2)$ is parallel to the line joining $(5, 1, 11) & (8, 2, 8)$

2. Three concurrent lines with direction cosines $(l_{i}, m_{i}, n_{i})$ , $i = 1, 2, 3$ are coplanar, if $|\begin{matrix} l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3} \end{matrix}| = 0$

3. The plane $x - 2 y + z = 21$ and the line $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 1}{3}$ are parallel.

Which of these is/are correct?

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Important Questions on Three Dimensional Geometry

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

Let

A B C

be a triangle whose circumcentre is at

P

. If the position vectors

A, B, C

and

P

are

\vec{a}, \vec{b}, \vec{c}

and

\frac{\vec{a} + \vec{b} + \vec{c}}{4}

respectively, then the position vector of the orthocentre of this triangle, is :

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

The point in the

x y -

plane which is equidistant from

(2, 0, 3), (0, 3, 2)

and

(0, 0, 1)

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

A line in the $3$ -dimensional space makes an angle $θ (0 < θ \leq \frac{π}{2})$ with both the $X$ and $Y -$ axes. Then, the set of all values of $θ$ is in the interval :

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

If a line makes an angle of

\frac{π}{4}

with the positive directions of each of

X

-axis and

Y

-axis, then the angle that the line makes with the positive direction of the

Z

axis is

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

The plane which bisects the line segment joining the points

(- 3, - 3, 4)

and

(3, 7, 6)

at right angles, passes through which one of the following points?

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

If a point

R (4, y, z)

lies on the line segment joining the points

P (2, - 3,4)

and

Q (8,0, 10),

then the distance of

R

from the origin is

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

What is the angle subtended by an edge of a regular tetrahedron at its center?

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

l, m, n

are the direction cosines of a line which makes angles

α, β

and

γ

with the coordinate axes

X, Y, Z

, respectively, then

l m + m n + n l

takes the maximum value when

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

Let

A B C

be an acute scalene triangle, and

O

and

H

be its circumcentre and orthocentre respectively. Further, let

N

be the midpoint of

O H .

The value of the vector sum

\vec{N A} + \vec{N B} + \vec{N C}

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

If the origin and the points

P (2, 3, 4), Q (1, 2, 3)

and

R (x, y, z)

are co-planar then

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

X Y -

plane divides the line joining the points

A (2, 3, - 5)

and

B (- 1, - 2, - 3)

in the ratio

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

If a variable plane, at a distance of

3

units from the origin, intersects the coordinate axes at

A, B & C

, then the locus of the centroid of

Δ A B C

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

A line makes angles

α, β, γ

with the coordinate axes. If

α + β = \frac{π}{2}

, then

{(\cos α + \cos β + \cos γ)}^{2}

is equal to

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

If a unit vector

\vec{a}

makes angles

\frac{π}{3}

with

\hat{i}, \frac{π}{4}

with

\hat{j}

and

θ \in (0, π)

with

\hat{k},

then a value of

θ

is:

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

The angle between the lines with direction ratios

(2, - 2, 1)

and

(1, - 2, 2)

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

The point

(1, - 3, 4)

lies in the octant

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

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

and

D (3, 5, 1)

are the vertices of a tetrahedron. If

E, F, G

are the centroids of its faces containing the point

A,

then the centroid of the triangle

E F G

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

If vector

\vec{r}

with direction cosine

l, m, n

is equally inclined to the co-ordinate axes, then the total number of such vectors is

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

A B C

is a triangle in a plane with vertices

A (2, 3, 5), B (- 1, 3, 2)

and

C (λ, 5, μ)

. If the median through

A

is equally inclined to the coordinate axes, then the value of

(λ^{3} + μ^{3} + 5)

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Direction Cosines and Direction Ratios

∆ A B C

has vertices at

A \equiv (2, 3, 5), B \equiv (- 1, 3, 2)

and

C \equiv (λ, 5, μ) .

If the median through

A

is equally inclined to the axes, then the values of

λ

and

μ

respectively are

Important Questions on Three Dimensional Geometry

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