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$P$ is the point of concurrency of three straight lines $L_{1}, L_{2}$ , and $L_{3}$ which are mutually perpendicular to each other. If one of them intersects the $x$ -axis and other intersects the $y$ -axis, while the third line passes through a fixed point $(0, 0, c)$ on the $z$ -axis, then which of the following options satisfy the locus of the point $P$ ?

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Important Questions on Vectors

MEDIUM

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

If points

P (4, 5, x), Q (3, y, 4)

and

R (5, 8, 0)

are collinear, then the value of

x + y

MEDIUM

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

If a line makes angles

120^{o}

and

60^{o}

with the positive directions of

X

and

Z

axes respectively then the angle made by the line with positive

Y -

axis is

HARD

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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 :

HARD

Mathematics>Coordinate Geometry>Vectors>Direction Angles, 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}

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

The point in the

x y -

plane which is equidistant from

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

and

(0, 0, 1)

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

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>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

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)

EASY

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

L

and

M

are two points with position vectors

2 \vec{a} - \vec{b}

and

\vec{a} + 2 \vec{b}

respectively. The position vector of the point

N

which divides the line segment

L M

in the ratio

2 : 1

externally is

MEDIUM

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

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

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

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>Vectors>Direction Angles, 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

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 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?

EASY

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

A variable plane passes through a fixed point

(3, 2, 1)

and meets

x, y

and

z

-axes at

A, B & C

respectively. A plane is drawn parallel to the

y z -

plane through

A

, a second plane is drawn parallel to the

z x -

plane through

B

and a third plane is drawn parallel to the

x y

- plane through

C

. Then the locus of the point of intersection of these three planes, is

Important Questions on Vectors

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