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Chapters
2: Inverse Trigonometric Functions
3: Matrices
4: Determinants
5: Continuity and Differentiability
6: Application of Derivatives
7: Integrals
8: Application of Integrals
9: Differential Equations
10: Vector Algebra
▶ 11: Three Dimensional Geometry
12: Linear Programming
13: Probability
![NCERT solutions for Mathematics [English] Class 12 chapter 11 - Three Dimensional Geometry - Shaalaa.com](/images/mathematics-english-class-12_6:f2fd4beccca84a5e862c6237e92b7e09.jpg "NCERT solutions for Mathematics [English] Class 12 chapter 11 - Three Dimensional Geometry")
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Solutions for Chapter 11: Three Dimensional Geometry
Below listed, you can find solutions for Chapter 11 of CBSE, Karnataka Board PUC NCERT for Mathematics [English] Class 12.
NCERT solutions for Mathematics [English] Class 12 11 Three Dimensional Geometry EXERCISE 11.1 [Page 381]
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Find the direction cosines of a line which makes equal angles with the coordinate axes.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).
NCERT solutions for Mathematics [English] Class 12 11 Three Dimensional Geometry EXERCISE 11.2 [Pages 389 - 390]
Show that the three lines with direction cosines `12/13, (-3)/13, (-4)/13; 4/13, 12/13, 3/13; 3/13, (-4)/13, 12/13 ` are mutually perpendicular.
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).
Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (−1, −2, 1), (1, 2, 5).
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector `3hati+2hatj-2hatk`.
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector `2hati -hatj+4hatk` and is in the direction `hati + 2hatj - hatk`.
Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and parallel to the line given by `(x+3)/3 = (y-4)/5 = (z+8)/6`.
The Cartesian equation of a line is `(x-5)/3 = (y+4)/7 = (z-6)/2` Write its vector form.
Find the angle between the following pair of lines:
`vecr = 2hati - 5hatj + hatk + lambda(3hati - 2hatj + 6hatk) and vecr = 7hati - 6hatk + mu(hati + 2hatj + 2hatk)`
Find the angle between the following pair of lines:
`vecr = 3hati + hatj - 2hatk + lambda(hati - hatj - 2hatk) and vecr = 2hati - hatj -56hatk + mu(3hati - 5hatj - 4hatk)`
Find the angle between the following pairs of lines:
`(x-2)/2 = (y-1)/5 = (z+3)/(-3)` and `(x+2)/(-1) = (y-4)/8 = (z -5)/4`
Find the angle between the following pairs of lines:
`x/y = y/2 = z/1` and `(x-5)/4 = (y-2)/1 = (z - 3)/8`
Find the values of p so the line `(1-x)/3 = (7y-14)/2p = (z-3)/2` and `(7-7x)/(3p) = (y -5)/1 = (6-z)/5` are at right angles.
Show that the lines `(x-5)/7 = (y + 2)/(-5) = z/1` and `x/1 = y/2 = z/3` are perpendicular to each other.
Find the shortest distance between the lines:
`vecr = (hati+2hatj+hatk) + lambda(hati-hatj+hatk)` and `vecr = 2hati - hatj - hatk + mu(2hati + hatj + 2hatk)`
Find the shortest distance between the lines.
`(x + 1)/7 = (y + 1)/(- 6) = (z + 1)/1` and `(x - 3)/1 = (y - 5)/(- 2) = (z - 7)/1`.
Find the shortest distance between the lines whose vector equations are `vecr = (hati + 2hatj + 3hatk) + lambda(hati - 3hatj + 2hatk)` and `vecr = 4hati + 5hatj + 6hatk + mu(2hati + 3hatj + hatk)`.
Find the shortest distance between the lines whose vector equations are `vecr = (1-t)hati + (t - 2)hatj + (3 -2t)hatk` and `vecr = (s+1)hati + (2s + 1)hatk`.
NCERT solutions for Mathematics [English] Class 12 11 Three Dimensional Geometry Miscellaneous Exercise [Pages 390 - 391]
Find the angle between the lines whose direction ratios are a, b, c and b − c, c − a, a − b.
Find the equation of a line parallel to x-axis and passing through the origin.
If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.
Find the shortest distance between lines `vecr = 6hati + 2hatj + 2hatk + lambda(hati - 2hatj + 2hatk)` and `vecr =-4hati - hatk + mu(3hati - 2hatj - 2hatk)`.
Find the vector equation of the line passing through the point (1, 2, − 4) and perpendicular to the two lines:
`(x -8)/3 = (y+19)/(-16) = (z - 10)/7 and (x - 15)/3 = (y - 29)/8 = (z- 5)/(-5)`
Solutions for 11: Three Dimensional Geometry
NCERT solutions for Mathematics [English] Class 12 chapter 11 - Three Dimensional Geometry
Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics [English] Class 12 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Mathematics [English] Class 12 CBSE, Karnataka Board PUC 11 (Three Dimensional Geometry) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
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Concepts covered in Mathematics [English] Class 12 chapter 11 Three Dimensional Geometry are Introduction of Three Dimensional Geometry, Angle Between Two Lines, Equation of a Plane in Normal Form, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Shortest Distance Between Two Lines, Equation of a Line in Space, Direction Cosines and Direction Ratios of a Line, Three - Dimensional Geometry Examples and Solutions, Equation of a Plane Passing Through Three Non Collinear Points, Relation Between Direction Ratio and Direction Cosines, Intercept Form of the Equation of a Plane, Coplanarity of Two Lines, Distance of a Point from a Plane, Angle Between Line and a Plane, Angle Between Two Planes, Vector and Cartesian Equation of a Plane, Distance of a Point from a Plane, Plane Passing Through the Intersection of Two Given Planes, Introduction of Three Dimensional Geometry, Angle Between Two Lines, Equation of a Plane in Normal Form, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Shortest Distance Between Two Lines, Equation of a Line in Space, Direction Cosines and Direction Ratios of a Line, Three - Dimensional Geometry Examples and Solutions, Equation of a Plane Passing Through Three Non Collinear Points, Relation Between Direction Ratio and Direction Cosines, Intercept Form of the Equation of a Plane, Coplanarity of Two Lines, Distance of a Point from a Plane, Angle Between Line and a Plane, Angle Between Two Planes, Vector and Cartesian Equation of a Plane, Distance of a Point from a Plane, Plane Passing Through the Intersection of Two Given Planes.
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