The equation of plane passing through the point P ( 1 , 1 , 1 ) and containing the line r → = ( − 3 i ^ + j ^ + 5 k ^ ) + λ ( 3 i ^ − j ^ − 5 k ^ ) would be. Also, show that the plane contains the line r → = ( − i ^ + 2 j ^ + 5 k ^ ) + μ ( i ^ − 2 j ^ − 5 k ^ ) .

r → ⋅ ( i ^ − 2 j ^ + k ^ ) = 0

r → ⋅ ( i ^ − 2 j ^ − k ^ ) = 0

r → ⋅ ( i ^ + 2 j ^ − k ^ ) = 0

r → ⋅ ( i ^ + 2 j ^ + k ^ ) = 0

The equation of plane passing through the point P ( 1 , 1 , 1 ) and containing the line r → = ( − 3 i ^ + j ^ + 5 k ^ ) + λ ( 3 i ^ − j ^ − 5 k ^ ) would be. Also, show that the plane contains the line r → = ( − i ^ + 2 j ^ + 5 k ^ ) + μ ( i ^ − 2 j ^ − 5 k ^ ) .

r → ⋅ ( i ^ − 2 j ^ + k ^ ) = 0

MEDIUM

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The equation of plane passing through the point $P (1, 1, 1)$ and containing the line $\vec{r} = (- 3 \hat{i} + \hat{j} + 5 \hat{k}) + λ (3 \hat{i} - \hat{j} - 5 \hat{k})$ would be.

Also, show that the plane contains the line $\vec{r} = (- \hat{i} + 2 \hat{j} + 5 \hat{k}) + μ (\hat{i} - 2 \hat{j} - 5 \hat{k}) .$

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Important Questions on Three Dimensional Geometry

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

L_{1}

is the line of intersection of the planes

2 x - 2 y + 3 z - 2 = 0, x - y + z + 1 = 0

and

L_{2}

is the line of intersection of the planes

x + 2 y - z - 3 = 0, 3 x - y + 2 z - 1 = 0,

then the distance of the origin from the plane, containing the lines

L_{1}

and

L_{2}

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The equation of the plane passing through the points

(1, 2, 3), (- 1, 4, 2)

and

(3, 1, 1)

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

Equation of the plane which passes through the point of intersection of lines

\frac{x - 1}{3} = \frac{y - 2}{1} = \frac{z - 3}{2}

and

\frac{x - 3}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}

and has the largest distance from the origin is:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The length of the perpendicular drawn from the point

(2, 1, 4)

to the plane containing the lines

\vec{r} = (\hat{i} + \hat{j}) + λ (\hat{i} + 2 \hat{j} - \hat{k})

and

\vec{r} = (\hat{i} + \hat{j}) + μ (- \hat{i} + \hat{j} - 2 \hat{k})

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The sum of the intercepts on the coordinate axes of the plane passing through the point

(- 2, - 2, 2)

and containing the line joining the points

(1, - 1, 2)

and

(1, 1, 1)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

A plane passing though the points

(0, - 1, 0)

and

(0, 0, 1)

and making an angle

\frac{π}{4}

with the plane

y - z + 5 = 0,

also passes through the point

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The coordinates of the foot of the perpendicular from the point

(1, - 2, 1)

on the plane containing the lines

\frac{x + 1}{6} = \frac{y - 1}{7} = \frac{z - 3}{8}

and

\frac{x - 1}{3} = \frac{y - 2}{5} = \frac{z - 3}{7},

is:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The plane which bisects the line joining the points

(4, - 2, 3)

and

(2, 4, - 1)

at right angles also passes through the point :

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The distance of the point

(1, 3, - 7)

from the plane passing through the point

(1, - 1, - 1)

, having normal perpendicular to both the lines

\frac{x - 1}{1} = \frac{y + 2}{- 2} = \frac{z - 4}{3}

and

\frac{x - 2}{2} = \frac{y + 1}{- 1} = \frac{z + 7}{- 1}

, is:

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The equation of the plane through

(- 1, 1, 2),

whose normal makes equal acute angle with co-ordinate axes is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

If the foot of the perpendicular drawn from the point

(0, 0, 0)

to the plane is

(4, - 2, - 5)

then the equation of the plane is …..

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The equation of the plane containing the straight line

\frac{x}{2} = \frac{y}{3} = \frac{z}{4}

and perpendicular to the plane containing the straight lines

\frac{x}{3} = \frac{y}{4} = \frac{z}{2}

and

\frac{x}{4} = \frac{y}{2} = \frac{z}{3}

is:

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

A plane passes through the point

(0, 1, 1)

and has normal vector

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

. Its equation is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The equation of the plane passing through the point

(1,1, 1)

and perpendicular to the planes

2 x + y - 2 z = 5

and

3 x - 6 y - 2 z = 7

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The sum of the intercepts made by the plane

\vec{r} \cdot (3 \hat{i} + \hat{j} + 2 \hat{k}) = 18

on the co-ordinate axes is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

If a variable plane in

3

-dimensional space moves in such a way that the sum of the reciprocals of its intercepts on the

x

and

y

-axes exceeds the reciprocal of its intercept on the

z

-axis by

2

, then all such planes will pass through the point

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The plane passing through the points

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

and parallel to the line,

2 x = 3 y

z = 1

also passes through the point

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

A plane is at a distance of

5

units from the origin and perpendicular to the vector

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

.

The equation of the plane is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The equation of the plane which bisects the line joining

(3, 0, 5)

and

(1, 2, - 1)

at right angles is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Equation of Plane

The plane passing through the point

(4, - 1, 2)

and parallel to the lines

\frac{x + 2}{3} = \frac{y - 2}{- 1} = \frac{z + 1}{2}

and

\frac{x - 2}{1} = \frac{y - 3}{2} = \frac{z - 4}{3}

also passes through the point

The equation of plane passing through the point P(1, 1, 1) and containing the line r→=(−3i^+j^+5k^)+λ(3i^−j^−5k^) would be. Also, show that the plane contains the line r → =( − i ^ +2 j ^ +5 k ^ )+μ( i ^ −2 j ^ −5 k ^ ).

Important Questions on Three Dimensional Geometry

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