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QUESTION: 1

Direction cosines of a line are

Solution:

Direction cosines of a line are the cosines of the angles made by the line with the positive direction of the coordinate axis.i.e. x- axis , y-axis and z – axis respectively.

QUESTION: 2

Shortest distance between two skew lines is

Solution:

Shortest distance between two skew lines is The line segment perpendicular to both the lines .

QUESTION: 3

Find the shortest distance between the lines

Solution:

On comparing the given equations with :

In the cartesian form two lines

we get ;

x_{1} = -1, y_{1} = -1,z_{1} = -1, ; a_{1} = 7, b_{1} = -6, c_{1} = 1 and

x_{2} = 3, y_{2} = 5, z_{2} = 7; a_{2} = 1, b_{2} = -2, c_{2} = 1

Now the shortest distance between the lines is given by :

QUESTION: 4

The angle θ between the planes A_{1}x + B_{1}y + C_{1}z + D1 = 0 and A_{2 }x + B_{2} y + C_{2} z + D_{2} = 0 is given by

Solution:

By definition , The angle θ between the planes A_{1}x + B_{1}y + C_{1}z + D_{1} = 0 and A_{2} x + B_{2} y + C_{2} z + D_{2} = 0 is given by :

QUESTION: 5

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

Solution:

The equation of the plane through the line of intersection of the planes

QUESTION: 6

If l, m, n are the direction cosines of a line, then

Solution:

If l, m , n are the direction cosines of a line then , we know that, l^{2}+ m^{2}+ n^{2 }= 1.

QUESTION: 7

Shortest distance between

Solution:

QUESTION: 8

Find the shortest distance between the lines :

Solution:

On comparing the given equations with:

, we get:

QUESTION: 9

The distance of a point whose position vector is from the plane

Solution:

The distance of a point whose position vector is from the plane given by :

QUESTION: 10

Find the angle between the planes whose vector equations are

Solution:

QUESTION: 11

is a vector joining two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}). If Direction cosines of are

Solution:

is a vector joining two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}). If Direction cosines of are given by :

QUESTION: 12

Shortest distance between the lines

Solution:

In Cartesian coordinate system Shortest distance between the lines

QUESTION: 13

Find the shortest distance between the lines and

Solution:

Find the shortest distance between the lines

On comparing them with :

we get :

QUESTION: 14

The distance d from a point P(x_{1}, y_{1}, z_{1}) to the plane Ax + By + Cz + D = 0 is

Solution:

The distance d from a point P(x_{1}, y_{1}, z_{1}) to the plane Ax + By + Cz + D = 0 is given by :

QUESTION: 15

Determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

Solution:

QUESTION: 16

If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are

Solution:

If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are Proportional to the direction cosines of the line.

QUESTION: 17

Distance between

Solution:

In vector form Distance between two parallel lines given by :

QUESTION: 18

Find the angle between the following pairs of lines: and

Solution:

If θ is the acute angle between

then cosine of the angle between

these two lines is given by :

Here,

Then,

QUESTION: 19

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane z = 2

Solution:

We have z = 2 . , it can be written as : 0x+0y+1z = 2. Compare it with lx+my+nz = d , we get ; l = 0 , m = 0 , n = 1 and d = 2 . therefore , D.C.’s of normal to the plane are 0 , 0 , 1 and distance from the origin = 2.

QUESTION: 20

In the following case, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them. 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

Solution:

We have ,

2x + y + 3z – 2 = 0 and x – 2y + 5 = 0. Let θ be the angle between the planes , then

QUESTION: 21

If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then

Solution:

If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then , the directions cosines of the line are given by :

QUESTION: 22

If a line makes angles 90^{∘}, 135^{∘}, 45^{∘} with the x, y and z – axes respectively, find its direction cosines.

Solution:

If a line makes angles 90^{∘}, 135^{∘}, 45^{∘} with the x, y and z – axes respectively, then the direction cosines of this line is given by :

QUESTION: 23

In the vector form, equation of a plane which is at a distance d from the origin, and is the unit vector normal to the plane through the origin is

Solution:

In the vector form, equation of a plane which is at a distance d from the origin, and is the unit vector normal to the plane through the origin is given by :

QUESTION: 24

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane x + y + z = 1

Solution:

Here , D.R’s of normal to the plane are 1, 1 , 1 ,its D.C ‘s are :

On dividing x + y + z = 1 by √3 , we get :

It is of the form : lx+my+nz = d , therefore , d = 1/√3 .

QUESTION: 25

In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

Solution:

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