EASY

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The equations of a line is 5x-3 = 15y +7 = 3-10z.Write the direction cosines of the line.

Important Questions on Three Dimensional Geometry

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

Let

A (3, 0, - 1), B (2, 10, 6)

and

C (1, 2, 1)

be the vertices of a triangle and

M

be the mid-point of

A C

. If

G

divides

B M

in the ratio,

2 : 1

, then

c o s (∠ G O A)

(

O

being the origin) is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The angle between the lines with direction ratios

(2, - 2, 1)

and

(1, - 2, 2)

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

∆ 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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

(a_{1}, b_{1}, c_{1}), (a_{2}, b_{2}, c_{2})

are the direction cosines of two lines making an angle

θ

with each other, then

\cos θ =

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The lines whose direction cosines are given by the relations

a l + b m + c n = 0

and

m n + n l + l m = 0

are

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The angle between the lines whose direction cosines satisfy the equations

l + m + n = 0

and

l^{2} = m^{2} + n^{2}

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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 :

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

If the angle between the lines

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

and

\frac{5 - x}{- 2} = \frac{7 y - 14}{P} = \frac{z - 3}{4}

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

then

P

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

A (3, 2, 3), B (1, 4, 6)

and

C (7, 4, 5)

are the three vertices of a parallelogram

A B C D

, then the angle between its diagonal through

D

and the side

D C

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The angle between the lines whose direction cosine satisfy the equations

l + m + n = 0

and

l^{2} = m^{2} + n^{2}

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The two lines

l x + m y = n

and

l^{'} x + m^{'} y = n^{'}

are perpendicular if

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

O \equiv (0, 0, 0), P \equiv (1, \sqrt{2}, 1),

then the acute angles made by the line

O P

with

X O Y

Y O Z, Z O X

planes are, respectively

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

If a line makes an angle of

\frac{π}{3}

with each of

x

and

y

-axis, then the acute angle made by

z

-axis is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

If lines

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

and

\frac{x - 1}{3 k} = \frac{y - 5}{1} = \frac{z - 6}{- 5}

are mutually perpendicular, then

k

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Basics of 3D Geometry

If the lines

x = a y + b, z = c y + d

and

x = a^{'} z + b^{'}, y = c^{'} z + d^{'}

are perpendicular, then

The equations of a line is 5x-3 = 15y +7 = 3-10z.Write the direction cosines of the line.

Important Questions on Three Dimensional Geometry

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