Three Dimensional Geometry Class 12th Notes - Free NCERT Class 12 Maths Chapter 11 Notes

Three Dimensional Geometry Class 12th Notes - Free NCERT Class 12 Maths Chapter 11 Notes - Download PDF

Edited By Ramraj Saini | Updated on Apr 23, 2022 01:13 PM IST

Notes for Class 12 Maths chapter 11 are regarding Three Dimensional Geometry. In chapter 11 we will be going through the geometric concepts in Three-dimensional geometry Class 12 notes. This Class 12 maths chapter 11 notes contains the following topics: direction cosines, direction ratios, equation of a straight line in vector and cartesian form, Equation of line passing through 2 points, the angle between them, coplanar, collinear and their conditions, equations in vector and cartesian forms, Equation of parallel and perpendicular lines and their distances, intercept form. NCERT class 12th Math chapter 11 notes contain standard formulas and detailed information that are implemented in problems. NCERT Class 12 Math chapter 11 notes and Class 12 Math chapter 12 notes contain a detailed explanation of topics, examples, exercises. The document will help students can cover all the topics that are in NCERT Notes for Class 12 Math chapter 11 textbook. It also contains the frequently asked questions by students which makes to know the concept of the topic. Every concept that is in CBSE Class 12 Maths chapter 11 notes is explained here in a simple way that students can get it easily. All these concepts can be downloaded as pdf from Class 12 Maths chapter 11 notes pdf download, ncert notes for Class 12 Maths chapter 11 are the notes for Three-dimensional geometry, Class 12 Three dimensional geometry notes pdf download.

Latest: JEE Main high scoring chapters | JEE Main 10 year's papers

This Story also Contains

NCERT Class 12 Chapter 11 Notes
Significance of NCERT Class 12 Maths Chapter 11 Notes:
Subject wise NCERT Exampler solutions
NCERT Books and Syllabus

Also, students can refer,

NCERT Class 12 Chapter 11 Notes

1646385805692

A line OA makes angles α,β,γ with x,y, and z-axis respectively. then cosα, cosβ, cosγ are called direction cosines.

l = cosα, m = cosβ, n = cosγ

Note: Direction cosines are always unique.

Direction ratios: The values that are proportional to directional cosines are called direction ratios.

$\\ \text{a,b,c are ratios, then}\ \frac{l}{a}=\frac{m}{b}\ =\frac{n}{c} \\ \\ \text{Direction cosines of ratios a,b,c :} \\ l = \pm \frac{a}{\sqrt{a^2+b^2+c^2}} \\ \\ m = \pm \frac{b}{\sqrt{a^2+b^2+c^2}} \\ \\ n = \pm \frac{c}{\sqrt{a^2+b^2+c^2}} \\ \\$

Direction cosines passing through two points:

Direction cosines:

Direction ratios:

NOTE: Directions ratios need not be unique

Equation of Stright Line:

Equation of a Line passing through the Point and parallel to vector b

$\\ where,\ \vec{a}\ \text{ is a position vector.}\ \vec{b}\ \text{ is a vector that is parallel to the line}$

Cartesian form:

where, be the point that line passes through and a, b, c are the direction ratios.

l,m,n are the direction cosines, then the equation is:

Equation of line that is passing through 2 points:

Points are :

are position vectors.

Cartesian form of two points:

Points are :

Vector equations of 2 lines:

$\vec{r_{1}}=\vec{a_{1}}+\lambda \left ( \vec{b_{1}}-\vec{a_{1}} \right ) and\ \vec{r_{2}}=\vec{a_{2}}+\lambda \left ( \vec{b_{2}}-\vec{a_{2}} \right )$

The angle between 2 lines:

Cartesian form: Let be an angle between the lines below:

Then,

1646387598149

Direction cosines ofwith angle are:

1646387597201

Few conditions:

when lines are perpendicular,

then cartesian form:

when lines are parallel

then cartesian form:

$\\ \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \\ \\ \frac{l_1}{l_2}=\frac{m_1}{m_2}=\frac{n_1}{n_2}$

Shortest path:

Vector equations of 2 lines:

$\vec{r_{1}}=\vec{a_{1}}+\lambda \vec{b_{1}}\ and\ \vec{r_{2}}=\vec{a_{2}}+\mu \vec{b_{2}}$

Shortest distance

1646388248135

$where\ \vec{a_1}, \vec{a_2}\ are\ position\ vectors\ and\ \vec{b_1}, \vec{b_2}\ \text{ are vectors in the direction of a line.}$

Cartesian form for two lines:

Cartesian form :

Shortest distance between the lines:

1646389059521

Distance between two Parallel Lines:

Lines are said to be coplanes when they are parallel.

Vector equations of 2 lines:

$\vec{r_{1}}=\vec{a_{1}} + \lambda \vec{b_{1}}\ and\ \vec{r_{2}}=\vec{a_{2}}+\mu \vec{b_{2}}$

Distance between two Parallel Lines:

1646389838805

Note: If lines are parallel, they always have the same direction ratios.

Distance between two points:

The midpoint of two points:

PLANE:

The plane is found unique, only if they satisfy any one of the following:

if normal form and distance from the origin are give
pass-through point and perpendicular
passes through non-collinear points

Equation of plane in normal form:

Cartesian form: one equation of plane is ax + by + cz = d and another equation of plane is lx + my + nz = p

Formula: Foot of perpendicular= (ld, md, nd).

Equation of plane perpendicular to vector and passing through a point:

Vector equation:

Cartesian form: Equation of plane that passes through the point

Equation of plane passing through non-collinear points:

Vector form:

Cartesian form: are non-collinear points

Equation: 1646390202442

If they are collinear: 1646390207371

Intercept Form:

Let a, b, c be x-intercept, y-intercept, z-intercept, respectively then

Equation of intercept= xa+yb+zc=1

Equation of Plane Passing through the intersection of two Planes:

equation of the planes is

$\vec{r_1}.\vec{n_1}=d_1 \ and \ \vec{r_2}.\vec{n_2}=d_2$

equation of the plane passing through the intersection :

Cartesian form: equation of planes are

equation of the plane passing through the intersection :

Coplanarity:

$\\ Lines:\vec{r_{1}}=\vec{a_{1}}+\lambda \vec{b_{1}}\ and\ \vec{r_{2}}=\vec{a_{2}}+\mu \vec{b_{2}}\ are\ coplanar\ then \\ Vector form: \left ( \vec{a_{1}}-\vec{a_{2}} \right )\cdot \left ( \vec{b_{1}}\times\vec{b_{2}} \right )=0$

Cartesian form:

For Two lines

are coplanar then

1646390202973

The angle between Two Planes: θ be the angle between two planes.

are normals, the angle between $\vec{r_1}.\vec{n_1}=d_1 \ and \ \vec{r_2}.\vec{n_2}=d_2$

1646390201824

Cartesian form: planes are ,

1646390203775

Angle Between Line and Plane:

Vector form: equation of a line is

angle θ between the line and the normal to the plane is

1646390201207

Cartesian form: a, b and c are direction ratios and lx + my + nz + d = 0 is equation of plane then

With this topic we conclude NCERT class 12 chapter 11 notes.

The link for the NCERT textbook pdf is given below:

URL: ncert.nic.in/ncerts/l/lemh205.pdf

Significance of NCERT Class 12 Maths Chapter 11 Notes:

NCERT Class 12 Maths chapter 11 notes are helpful for students to understand the topics well. In Three Dimensional Geometry Class 12 chapter 10 notes we have discussed many topics: direction cosines, direction ratios, equation of a straight line in vector and cartesian form, Equation of line passing through 2 points, angle between them, coplanar, collinear and their conditions, equations in vector and cartesian forms, Equation of parallel and perpendicular lines and their distances, intercept form. NCERT Class 12 Mathematics chapter 12 is also very useful and covers major topics of Class 12 CBSE Mathematics Syllabus.

The CBSE Class 12 Maths chapter 12 will help to understand the formulas, statements, and topics. There are also most frequently asked questions along with topic wise explanations. By referring to the document, gives the knowledge on all the topics of Class 12 chapter 11 Three Dimensional Geometry pdf download.

NCERT Class 12 Notes Chapter Wise.

NCERT Class 12 Maths Chapter 1 Notes

NCERT Class 12 Maths Chapter 2 Notes

NCERT Class 12 MathsChapter 3 Notes

NCERT Class 12 Maths Chapter 4 Notes

NCERT Class 12 Maths Chapter 5 Notes

NCERT Class 12 Maths Chapter 6 Notes

NCERT Class 12 Maths Chapter 7 Notes

NCERT Class 12 Maths Chapter 8 Notes

NCERT Class 12 Maths Chapter 9 Notes

NCERT Class 12 Maths Chapter 10 Notes

NCERT Class 12 Maths Chapter 11 Notes

NCERT Class 12 Maths Chapter 12 Notes

NCERT Class 12 Maths Chapter 13 Notes

Subject wise NCERT Exampler solutions

Subject wise NCERT solutions

JEE Main Highest Scoring Chapters & Topics

Just Study 40% Syllabus and Score upto 100%

Download EBook

NCERT Books and Syllabus

NCERT Book for Class 12

NCERT Syllabus for Class 12

Also Read

NCERT Class 12 Maths Notes - Download Chapter wise Free PDFs

NCERT Class 12 Physics Chapter 9 Notes Ray Optics and Optical Instruments - Download PDF

NCERT Class 12 Physics Chapter 8 Notes Electromagnetic Waves - Download PDF

NCERT Class 12 Physics Chapter 7 Notes Alternating Current - Download PDF

NCERT Class 12 Physics Chapter 6 Notes Electromagnetic Induction - Download PDF

NCERT Class 12 Physics Chapter 5 Notes Magnetism and Matter - Download PDF

NCERT Books for Class 12 - Download All Subjects PDFs Here

NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals

NCERT Solutions for Exercise 4.1 Class 11 Maths Chapter 4 - Principle of Mathematical Induction

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Exemplar Class 12 Physics Solutions Chapter 8 Electromagnetic Waves

Articles

UP Board Practical Exams 2025 - UPMSP High School and Inter Viva to Be Recorded to Ensure Transparency

Dec 18, 2024

MP Board 12th Model Paper 2024-25- Download English & Hindi Medium PDF Here

Dec 18, 2024

CBSE Class 11 Date Sheet 2025 – Download Class 11th Exam Schedule for All Subjects

Dec 18, 2024

Upcoming School Exams

National Institute of Open Schooling 10th examination

Application Date:21 November,2024 - 20 December,2024

Application Process

Exam Pattern

Mock Test

National Institute of Open Schooling 12th Examination

Application Date:21 November,2024 - 20 December,2024

Application Process

Exam Pattern

Admit Card

National Means Cum-Merit Scholarship

Admit Card Date:27 November,2024 - 22 December,2024

Eligibility Criteria

Application Process

Exam Pattern

West Bengal Board 12th Examination

Exam Date:02 December,2024 - 20 December,2024

Result

Exam Pattern

Admit Card

Gujarat Board Secondary School Certificate Examination

Late Fee Application Date:11 December,2024 - 20 December,2024

Exam Pattern

Admit Card

Result

View All School Exams

Get answers from students and experts

Popular Questions

A block of mass 0.50 kg is moving with a speed of 2.00 ms^-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up ? Fat supplies 3.8×10⁷J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take g = 9.8 ms⁻² :

Option 1)

2.45×10⁻³ kg

Option 2)

6.45×10⁻³ kg

Option 3)

9.89×10⁻³ kg

Option 4)

12.89×10⁻³ kg

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

A particle is projected at 60⁰ to the horizontal with a kinetic energy $K$ . The kinetic energy at the highest point

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

In the reaction,

$2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, $Mg_{3}(PO_{4})_{2}$ will contain 0.25 mole of oxygen atoms?

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol^-1) is

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t²) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m² , the number of rotations made by the pulley before its direction of motion if reversed, is