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QUESTION: 1

Skew lines are lines in different planes which are

Solution:

By definition : The Skew lines are lines in different planes which are neither parallel nor intersecting .

QUESTION: 2

If a line has the direction ratios – 18, 12, – 4, then what are its direction cosines ?

Solution:

If a line has the direction ratios – 18, 12, – 4, then its direction cosines are given by:

QUESTION: 3

Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is.

Solution:

In Cartesian co – ordinate system Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is given by : lx + my + nz = d.

QUESTION: 4

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

Solution:

QUESTION: 5

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 – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0

Solution:

QUESTION: 6

Angle between skew lines is

Solution:

Angle between skew lines is the angle between two intersecting lines drawn from any point

QUESTION: 7

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector

Solution:

The equation of the line which passes through the point (1, 2, 3) and is parallel to the vector Let vector and vector

QUESTION: 8

The equation of a plane through a point whose position vector is perpendicular to the vector . is

Solution:

In vector form The equation of a plane through a point whose position vector is perpendicular to the vector Is given by :

QUESTION: 9

Find the Cartesian equation of the plane

Solution:

QUESTION: 10

In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 4x + 8y + z – 8 = 0 and y + z – 4 = 0

Solution:

We have, 4x + 8y + z – 8 = 0 and y + z – 4 = 0. Let be the angle between the planes, then

QUESTION: 11

If l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are the direction cosines of two lines; and θ is the acute angle between the two lines; then

Solution:

If l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are the direction cosines of two lines; and θ is the acute angle between the two lines; then the cosine of the angle between these two lines is given by :

QUESTION: 12

Find the equation of the line in cartesian form that passes through the point with position vector and is in the direction

Solution:

, then , its Cartesian equation is given by :

QUESTION: 13

Equation of a plane passing through three non collinear points (x_{1}, y_{1}, z_{1}),(x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is

Solution:

In cartesian co – ordinate system : Equation of a plane passing through three non collinear points (x_{1}, y_{1}, z_{1}),(x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is given

QUESTION: 14

Find the Cartesian equation of the plane

Solution:

QUESTION: 15

Find the distance of the point (0, 0, 0) from the plane 3x – 4y + 12 z = 3

Solution:

As we know that the length of the perpendicular from point

P(x_{1},y_{1},z_{1}) from the plane a_{1}x+b_{1}y+c_{1}z+d_{1} = 0 is given by:

QUESTION: 16

If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are the direction ratios of two lines and θ is the acute angle between the two lines; then

Solution:

If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are the direction ratios of two lines and θθ is the acute angle between the two lines; then , the cosine of the angle between these two lines is given by :

QUESTION: 17

Find the equation of the line in cartesian form that passes through the point (– 2, 4, – 5) and parallel to the line given by

Solution:

Find the equation of the line in cartesian form that passes through the point (– 2, 4, – 5) and parallel to the line given by

is given by:

And l = 3 , m = 5 and n = 6 .

QUESTION: 18

Vector equation of a plane that contains three non collinear points having position vectors

Solution:

Vector equation of a plane that contains three non collinear points having position vectors

QUESTION: 19

The vector and cartesian equations of the planes that passes through the point (1, 0, – 2) and the normal to the plane is

Solution:

Let

be the position vector of the point here,

. Therefore, the required vector equation of the plane is:

QUESTION: 20

Find the distance of the point (3, – 2, 1) from the plane 2x – y + 2z + 3 = 0

Solution:

As we know that the length of the perpendicular from point P(x_{1},y_{1},z_{1}) from the plane

Here, P(3, - 2,1) is the point and equation of Plane is 2x - y + 2z+3 = 0

Therefore, the perpendicular distance is :

QUESTION: 21

Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector is

Solution:

Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector is given by :

QUESTION: 22

Find the values of p so that the lines are at right angles.

Solution:

Give lines are :

and

The D.R's of the lines are -3, 2p/7, 2 and -3p/7, 1, -5

QUESTION: 23

Vector equation of a plane that passes through the intersection of planes expressed in terms of a non – zero constant λ is

Solution:

In vector form:

Vector equation of a plane that passes through the intersection of planes

expressed in terms of a non – zero constant λ is given by :

QUESTION: 24

Find the equations of the planes that passes through three points (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)

Solution:

In cartesian co-ordinate system :

Equation of a plane passing through three non collinear

Points (x_{1}, y_{1}, z_{1}) , (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is given by :

Therefore, the equations of the planes that passes through three points (1,1,0), (1,2,1), (-2,2,-1) is given by :

⇒ (x-1)(-2) - (y-1) (3) + 3z = 0

⇒ 2x+3y - 3z = 5

QUESTION: 25

Find the distance of the point (2, 3, – 5) from the plane x + 2y – 2z = 9

Solution:

As we know that the length of the perpendicular from point P(x_{1},y_{1},z_{1}) from the plane

Here, P(2,3,-5) is the point and equation of plane is x+2y - 2z = 9

Therefore, the perpendicular distance is :

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