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

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BoardCBSE
TextbookNCERT
ClassClass 12
SubjectMaths
ChapterChapter 11
Chapter NameThree Dimensional Geometry
ExerciseEx 11.2
Number of Questions Solved17
CategoryNCERT Solutions

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

Ex 11.2 Class 12 Maths Question 1.
Show that the three lines with direction cosines:
frac { 12 }{ 13 } ,frac { -3 }{ 13 } ,frac { -4 }{ 13 } ,frac { 4 }{ 13 } ,frac { 12 }{ 13 } ,frac { 3 }{ 13 } ,frac { 3 }{ 13 } ,frac { -4 }{ 13 } ,frac { 12 }{ 13 }
are mutually perpendicular.
Solution:
Let the lines be L1,L2 and L3.
∴ For lines L1 and L2
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 1
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 1.1

Ex 11.2 Class 12 Maths 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
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 2

Ex 11.2 Class 12 Maths 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
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 3

Ex 11.2 Class 12 Maths Question 4.
Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector 3hat { i } +2hat { j } -2hat { k }
Solution:
Equation of the line passing through the point
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 4

Ex 11.2 Class 12 Maths Question 5.
Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2hat { i } -hat { j } +4hat { k } and is in the direction hat { i } +2hat { j } -hat { k } .
Solution:
The vector equation of a line passing through a point with position vector overrightarrow { a } and parallel to the
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 5
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 5.1

Ex 11.2 Class 12 Maths 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 frac { x+3 }{ 3 } =frac { y-4 }{ 5 } =frac { z+8 }{ 6 }
Solution:
The cartesian equation of the line passing through the point (-2,4, -5) and parallel to the
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 6

Ex 11.2 Class 12 Maths Question 7.
The cartesian equation of a line is
frac { x-5 }{ 3 } =frac { y+4 }{ 7 } =frac { z-6 }{ 2 }
write its vector form.
Solution:
The cartesian equation of the line is
frac { x-5 }{ 3 } =frac { y+4 }{ 7 } =frac { z-6 }{ 2 }
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
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 7

Ex 11.2 Class 12 Maths 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
therefore overrightarrow { a } =overrightarrow { 0 }
Direction ratios of the line passing through the
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 8
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 8.1

Ex 11.2 Class 12 Maths 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)
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 9

Ex 11.2 Class 12 Maths Question 10.
Find the angle between the following pair of lines
(i) overrightarrow { r } =2hat { i } -5hat { j } +hat { k } +lambda (3hat { i } +2hat { j } +6hat { k } )
andquad overrightarrow { r } =7hat { i } -6hat { j } +mu (hat { i } +2hat { j } +2hat { k } )
(ii) overrightarrow { r } =3hat { i } +hat { j } -2hat { k } +lambda (hat { i } -hat { j } -2hat { k } )
overrightarrow { r } =2hat { i } -hat { j } -56hat { k } +mu (3hat { i } -5hat { j } -4hat { k } )
Solution:
(i) Let θ be the angle between the given lines.
The given lines are parallel to the vectors
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 10
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 10.1

Ex 11.2 Class 12 Maths Question 11.
Find the angle between the following pair of lines
(i) frac { x-2 }{ 2 } =frac { y-1 }{ 5 } =frac { z+3 }{ -3 } andfrac { x+2 }{ -1 } =frac { y-4 }{ 8 } =frac { z-5 }{ 4 }
(ii) frac { x }{ 2 } =frac { y }{ 2 } =frac { z }{ 1 } andfrac { x-5 }{ 4 } =frac { y-2 }{ 1 } =frac { z-3 }{ 8 }
Solution:
Given
(i) frac { x-2 }{ 2 } =frac { y-1 }{ 5 } =frac { z+3 }{ -3 } andfrac { x+2 }{ -1 } =frac { y-4 }{ 8 } =frac { z-5 }{ 4 }
(ii) frac { x }{ 2 } =frac { y }{ 2 } =frac { z }{ 1 } andfrac { x-5 }{ 4 } =frac { y-2 }{ 1 } =frac { z-3 }{ 8 }
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 11

Ex 11.2 Class 12 Maths Question 12.
Find the values of p so that the lines
frac { 1-x }{ 3 } =frac { 7y-14 }{ 2p } =frac { z-3 }{ 2 } andfrac { 7-7x }{ 3p } =frac { y-5 }{ 1 } =frac { 6-z }{ 5 }
are at right angles
Solution:
The given equation are not in the standard form
The equation of given lines is
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 12

Ex 11.2 Class 12 Maths Question 13.
Show that the lines frac { x-5 }{ 7 } =frac { y+2 }{ -5 } =frac { z }{ 1 } andfrac { x }{ 1 } =frac { y }{ 2 } =frac { z }{ 3 } are perpendicular to each other
Solution:
Given lines
frac { x-5 }{ 7 } =frac { y+2 }{ -5 } =frac { z }{ 1 } …(i)
frac { x }{ 1 } =frac { y }{ 2 } =frac { z }{ 3 } …(ii)
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 13

Ex 11.2 Class 12 Maths Question 14.
Find the shortest distance between the lines
overrightarrow { r } =(hat { i } +2hat { j } +hat { k } )+lambda (hat { i } -hat { j } +hat { k } ) and
overrightarrow { r } =(2hat { i } -hat { j } -hat { k } )+mu (2hat { i } +hat { j } +2hat { k } )
Solution:
The shortest distance between the lines
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 14
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 14.1

Ex 11.2 Class 12 Maths Question 15.
Find the shortest distance between the lines
frac { x+1 }{ 7 } =frac { y+1 }{ -6 } =frac { z+1 }{ 1 } andfrac { x-3 }{ 1 } =frac { y-5 }{ -2 } =frac { z-7 }{ 1 }
Solution:
Shortest distance between the lines
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 15

Ex 11.2 Class 12 Maths Question 16.
Find the distance between die lines whose vector equations are:
overrightarrow { r } =(hat { i } +2hat { j } +3hat { k) } +lambda (hat { i } -3hat { j } +2hat { k } ) and
overrightarrow { r } =(4hat { i } +5hat { j } +6hat { k) } +mu (2hat { i } +3hat { j } +hat { k } )
Solution:
Comparing the given equations with
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 16
tiwari academy class 12 maths Chapter 11 Three Dimensional Geometry 16.1

Ex 11.2 Class 12 Maths Question 17.
Find the shortest distance between the lines whose vector equations are
overrightarrow { r } =(1-t)hat { i } +(t-2)hat { j } +(3-2t)hat { k } and
overrightarrow { r } =(s+1)hat { i } +(2s-1)hat { j } -(2s+1)hat { k }
Solution:
Comparing these equation with
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry 17

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