NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2.

• Three Dimensional Geometry Class 12 Ex 11.1
• Three Dimensional Geometry Class 12 Ex 11.3

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 11
Chapter Name	Three Dimensional Geometry
Exercise	Ex 11.2
Number of Questions Solved	17
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2

Question 1.
Show that the three lines with direction cosines:

are mutually perpendicular.
Solution:
Let the lines be L1,L2 and L3.
∴ For lines L1 and L2

Question 2.
Show that the line through the points (1,-1,2) (3,4, -2) is perpendicular to the line through the points (0,3,2) and (3,5,6).
Solution:
Let A, B be the points (1, -1, 2), (3, 4, -2) respectively Direction ratios of AB are 2,5, -4
Let C, D be the points (0, 3, 2) and (3, 5, 6) respectively Direction ratios of CD are 3, 2,4 AB is Perpendicular to CD if

Question 3.
Show that the line through the points (4,7,8) (2,3,4) is parallel to the line through the points (-1,-2,1) and (1,2,5).
Solution:
Let the points be A(4,7,8), B (2,3,4), C (-1,-2,1) andD(1,2,5).
Now direction ratios of AB are

Question 4.
Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector
Solution:
Equation of the line passing through the point

Question 5.
Find the equation of the line in vector and in cartesian form that passes through the point with position vector and is in the direction .
Solution:
The vector equation of a line passing through a point with position vector and parallel to the

Question 6.
Find the cartesian equation of the line which passes through the point (-2,4, -5) and parallel to the line is given by
Solution:
The cartesian equation of the line passing through the point (-2,4, -5) and parallel to the

Question 7.
The cartesian equation of a line is

write its vector form.
Solution:
The cartesian equation of the line is

Clearly (i) passes through the point (5, – 4, 6) and has 3,7,2 as its direction ratios.
=> Line (i) passes through the point A with

Question 8.
Find the vector and the cartesian equations of the lines that passes through the origin and (5,-2,3).
Solution:
The line passes through point

Direction ratios of the line passing through the

Question 9.
Find the vector and cartesian equations of the line that passes through the points (3, -2, -5), (3,-2,6).
Solution:
The PQ passes through the point P(3, -2, -5)

Question 10.
Find the angle between the following pair of lines
(i)

(ii)

Solution:
(i) Let θ be the angle between the given lines.
The given lines are parallel to the vectors

Question 11.
Find the angle between the following pair of lines
(i)
(ii)
Solution:
Given
(i)
(ii)

Question 12.
Find the values of p so that the lines

are at right angles
Solution:
The given equation are not in the standard form
The equation of given lines is

Question 13.
Show that the lines are perpendicular to each other
Solution:
Given lines
…(i)
…(ii)

Question 14.
Find the shortest distance between the lines
and

Solution:
The shortest distance between the lines

Question 15.
Find the shortest distance between the lines

Solution:
Shortest distance between the lines

Question 16.
Find the distance between die lines whose vector equations are:
and

Solution:
Comparing the given equations with

Question 17.
Find the shortest distance between the lines whose vector equations are
and

Solution:
Comparing these equation with

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