If x -coordinate of a point P on the line joining the points Q 2 , 2 , 1 and R 5 , 2 , - 2 is 4 , then the y -coordinate of P =

Sum of x and z coordinates of P

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If the point $(a, 8, - 2)$ divides the line segment joining the points $(1, 4, 6)$ and $(5, 2, 10)$ in the ratio $m : n$ then $\frac{2 m}{n} - \frac{a}{3} =$

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Important Questions on Three Dimensional Geometry

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

x

-coordinate of a point

P

on the line joining the points

Q (2, 2, 1)

and

R (5, 2, - 2)

4

, then the

y

-coordinate of

P =

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

If the mid points of the sides

A B, B C

and

C A

of a triangle are respectively

D (1, 2, - 3),

E (3, 0, 1)

and

F (- 1, 1, - 4),

then the centroid of the triangle

A D F

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

Let the position vectors of two points

P

and

Q

2 \hat{i} + \hat{j}

and

\hat{i} - 3 \hat{j}

respectively. Then the position vector of a point

R

which divides the line joining the points

P

and

Q

in the ratio

1 : 2

externally, is

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

If the vectors

\vec{A B} = - 3 \hat{i} + 4 \hat{k}

and

\vec{A C} = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}

are the sides of a triangle

A B C

. Then the length of the median through

A

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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>Three Dimensional Geometry>Basics of 3D Geometry

The centroid of a triangle with vertices

A (3, 4, 5), B (6, 7, 2)

and

C (x, y, z)

(3, 2, 3)

then

x + y + z =

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The position vectors of the points

A

and

B

with respect to

O

are

2 \hat{i} + 2 \hat{j} + \hat{k}

and

2 \hat{i} + 4 \hat{j} + 4 \hat{k}

. The length of the internal bisector of

∠ B O A

∆ A O B

is (take proportionality constant is

2

)

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

What is the ratio in which the line segment joining the points

(- 3, 2, 5)

and

(6, 3, 2)

is divided by the

X Y

- plane?

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The mid-points of the sides of triangle are

(1, 5, - 1), (0, 4, - 2)

and

(2, 3, 4)

, then centroid of the triangle is

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

X Y -

plane divides the line joining the points

A (2, 3, - 5)

and

B (- 1, - 2, - 3)

in the ratio

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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>Basics of 3D Geometry

In a triangle

A B C,

if the mid points of sides

A B, B C, C A

are

(3, 0, 0), (0, 4, 0), (0, 0, 5)

, respectively, then

A B^{2} + B C^{2} + C A^{2} =

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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?

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

Let

A

and

B

be two points with position vectors

\vec{a}

and

\vec{b}

respectively and let

C

be a point dividing

A B

internally and the position vector of

C

A B

\vec{c} = λ \vec{a} + μ \vec{b},

then

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The position vectors of the points

P

and

Q

are respectively

- 2 \bar{i} - 3 \bar{j} + \bar{k}

and

3 \bar{i} + 3 \bar{j} + 2 \bar{k} .

The ratio in which the point having position vector

\frac{- 9}{2} \bar{i} - 6 \bar{j} + \frac{1}{2} \bar{k}

divides the line segment joining

P

and

Q

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Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

If the vertices of the triangles are

(1, 2, 3), (2, 3, 1), (3, 1, 2)

and if

H, G, S

and

I

respectively denote its orthocenter, centroid, circumcenter and incenter, then

H + G + S + I =

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

The ratio in which the

Y Z -

plane divides the line joining

(2, 4, 5)

and

(3, 5, - 4)

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Basics of 3D Geometry

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

If the point a, 8,-2 divides the line segment joining the points 1,4,6 and 5,2,10 in the ratio m:n then 2mn-a3=

Important Questions on Three Dimensional Geometry

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If the point $(a, 8, - 2)$ divides the line segment joining the points $(1, 4, 6)$ and $(5, 2, 10)$ in the ratio $m : n$ then $\frac{2 m}{n} - \frac{a}{3} =$