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

EASY

Earn 100

Find the volume and surface area of a regular tetrahedron with side $a = 7 cm$ .

Important Questions on Three Dimensional Coordinates

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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} =

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

X Y -

plane divides the line joining the points

A (2, 3, - 5)

and

B (- 1, - 2, - 3)

in the ratio

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

Find

a, b, c

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

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

The distance of point

P (3, 4, 5)

from the

y z

-plane is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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} =

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

Three lines

L_{1} : \vec{r} = λ \hat{i}, λ \in R,

L_{2} : \vec{r} = \hat{k} + μ \hat{j}, μ \in R

and

L_{3} : \vec{r} = \hat{i} + \hat{j} + ν \hat{k}, ν \in R

are given. For which point(s)

Q

and

L_{2}

can we find a point

P

L_{1}

and a point

R

L_{3}

so that

P, Q

and

R

are collinear?

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

The point

(4, 5, - 6)

lies in

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

(\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>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

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

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

If a plane cuts off intercepts

O A = a, O B = b, O C = c

from the co-ordinate axes, then the area of the triangle

A B C =

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Coordinates>Distance and Section Formula in 3D

If the sum of the squares of the distance of a point from the three coordinate axes be

36

, then its distance from the origin is

Find the volume and surface area of a regular tetrahedron with side a=7 cm.

Important Questions on Three Dimensional Coordinates

Learn and Prepare for any exam you want !

Find the volume and surface area of a regular tetrahedron with side $a = 7 cm$ .