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If $(2, 3, c)$ are the direction ratios of a ray passing through the point $C (5, q, 1)$ and also the mid point of the line segment joining the points $A (p, - 4.2)$ and $B (3, 2, - 4)$ then $c . (p + 7 q) =$

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Important Questions on Vectors

EASY

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

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

The angle between the lines with direction ratios

(2, - 2, 1)

and

(1, - 2, 2)

MEDIUM

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

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

EASY

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

A line making angles

45 °

and

60 °

with positive directions of the axis of

x

and

y

makes with the positive direction of

z

-axis, an angle of

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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)

MEDIUM

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

If a line makes angles

\tan^{- 1} \sqrt{7}, \tan^{- 1} (\sqrt{\frac{5}{3}})

with

X

-axis and

Y

-axis respectively, then the angle made by it with

Z

-axis is

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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

MEDIUM

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

A straight line makes an angle of

30 °, 45 °

and

60 °

with the positive direction of

X -

axis,

Y -

axis and

Z -

axis respectively. What are the direction cosines of the straight line?

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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:

EASY

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

The direction angles of the line

x = 4 z + 3, y = 2 - 3 z

are

α, β

and

γ,

then

\cos α + \cos β + \cos γ =

_______.

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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

MEDIUM

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

The number of straight lines that are equally inclined to the three-dimensional coordinate axes is

EASY

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

If a line in octant

O X Y Z

makes equal angles with co-ordinate axes, then

MEDIUM

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

α, β, γ

are the angles made by a vector with the co-ordinate axes, then

\sin^{2} α + \sin^{2} β + \sin^{2} γ =

EASY

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

A vector makes equal angles

α

with

x

and

y

axes and

90 °

with

z -

axis. Then

α =

HARD

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

The distance of the point

(1, - 2, 3)

from the plane

x - y + z = 5

measured parallel to a line, whose direction ratios are

2, 3, - 6

, is

EASY

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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

HARD

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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>Vectors>Direction Angles, 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

If 2,3,c are the direction ratios of a ray passing through the point C5,q,1 and also the mid point of the line segment joining the points Ap,-4.2 and B3,2,-4 then c.p+7q=

Important Questions on Vectors

Learn and Prepare for any exam you want !

If $(2, 3, c)$ are the direction ratios of a ray passing through the point $C (5, q, 1)$ and also the mid point of the line segment joining the points $A (p, - 4.2)$ and $B (3, 2, - 4)$ then $c . (p + 7 q) =$