Find a , b , c if a 1 , 3 , 2 + b 1 , - 5 , 6 + c 2 , 1 , - 2 = 4 , 10 , - 8 .

Given a 1 , 3 , 2 + b 1 , - 5 , 6 + c 2 , 1 , - 2 = 4 , 10 , - 8 ⇒ ( a + b + 2 c , 3 a - 5 b + c , 2 a + 6 b - 2 c ) = ( 4 , 10 , - 8 ) a + b + 2 c = 4 . . . . . . i 3 a - 5 b + c = 10 . . . . . ii 2 a + 6 b - 2 c = - 8 . . . . . . . iii Adding i and iii equations, 3 a + 7 b = - 4 . . . . . iv Multiply equation ii by 2 and add with equation iii , 6 a - 10 b + 2 c + 2 a + 6 b - 2 c = 20 - 8 ⇒ 8 a - 4 b = 12 . . . . . v Now, 8 × equation iv - 3 × equation v , ⇒ 24 a + 56 b - 24 a + 12 b = - 32 - 36 ⇒ 68 b = - 68 ⇒ b = - 1 From equation iv , 3 a + 7 - 1 = - 4 ⇒ 3 a = 3 ⇒ a = 1 From equation i , 1 - 1 + 2 c = 4 ⇒ c = 2 Therefore, a = 1 , b = - 1 , c = 2

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The point $(4, 5, - 6)$ lies in

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Important Questions on Vectors

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

Let

A (1, 2, 3), B (- 1, 4, 6), C (0, - 6, 4)

and

D (1, 1, 1)

be the vertices of a tetrahedron,

G

be its centroid and

G_{1}

be the centroid of its face

B C D

. Then

\frac{A G_{1}}{A G} =

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

(\frac{9}{4}, \frac{5}{4}, \frac{15}{4})

is the centroid of a tetrahedron whose vertices are

(a, 2, 1), (1, b, 4),

(4, 0, c)

and

(1, 1, 7),

then

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

Find

a, b, c

a (1, 3, 2) + b (1, - 5, 6) + c (2, 1, - 2) = (4, 10, - 8)

HARD

Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

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}

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

The distance of point

P (3, 4, 5)

from the

y z

-plane is

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

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?

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

If the line

O P

of length

r

makes an angle

α

with

x

-axis and lies in the

X Z

-plane, then coordinates of

P

are

HARD

Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

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

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

The octant in which the point

(2, - 4, - 7)

lies is

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

The position vector of point

A

(4, 2, - 3) .

p_{1}

is perpendicular distance of

A

from

X Y

-plane and

p_{2}

is perpendicular distance from

Y

-axis, then

p_{1} + p_{2} =

_______.

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

If the origin and the points

P (2, 3, 4), Q (1, 2, 3)

and

R (x, y, z)

are co-planar then

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

The graph of the equation

x^{2} + y^{2} = 0

in three-dimensional space is

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

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

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

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

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

∆ 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

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

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 :

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Mathematics>Coordinate Geometry>Vectors>Basics of Three Dimensional Coordinate System

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 :