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Consider two lines whose direction ratios are $(2, - 1, 2)$ and $(k, 3, 5)$ . They are inclined at an angle $\frac{π}{4}$ .

What are the direction ratios of a line which is perpendicular to both the lines?

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Important Questions on Vectors

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

The angle between the line

A B

and

C D

, where

A = (0, 0, 0), B = (1, 1, 1), C = (- 1, - 1, - 1)

and

D = (0, 1, 0)

is given by

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

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

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

If the lines

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

and

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

are coplanar, then

k

can have

HARD

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

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

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

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

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

The angle between the two diagonals of a cube is

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

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 .

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

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>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

Two lines

L_{1} : x = 5, \frac{y}{3 - α} = \frac{z}{- 2}

and

L_{2} : x = α, \frac{y}{- 1} = \frac{z}{2 - α}

are coplanar. Then

α

can take value(s)

HARD

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

The angle between the lines whose direction cosines satisfy the equations

l + m + n = 0

and

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

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

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

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

If the lines

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

and

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

are coplanar, then

k

can have

HARD

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

The angle between the lines whose direction cosine satisfy the equations

l + m + n = 0

and

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

HARD

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

If for some,

α \in R,

the lines

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

and

L_{2} : \frac{x + 2}{α} = \frac{y + 1}{5 - α} = \frac{z + 1}{1}

are coplanar, then the line

L_{2}

passes through the point :

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

(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>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

The angle between a line with direction ratios

2, 2, 1

and the line joining the points

(3, 1, 4)

and

(7, 2, 12)

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

The two lines

l x + m y = n

and

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

are perpendicular if

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

The number of distinct real values of

λ

, for which the lines

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

and

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

, are coplanar is

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

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>Coordinate Geometry>Vectors>Direction Angles, Cosines and Direction Ratios

If the lines

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

and

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

are perpendicular, then

Consider two lines whose direction ratios are 2, -1, 2 and k, 3, 5. They are inclined at an angle π4. What are the direction ratios of a line which is perpendicular to both the lines?

Important Questions on Vectors

Learn and Prepare for any exam you want !

Consider two lines whose direction ratios are $(2, - 1, 2)$ and $(k, 3, 5)$ . They are inclined at an angle $\frac{π}{4}$ .

What are the direction ratios of a line which is perpendicular to both the lines?