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.

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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

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