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If you have been asked the position of a butterfly flying in space around at an instant of time, what distance should you measure from a reference point? To answer this, you need to understand the concept of three-dimensional geometry. Three-dimensional geometry talks about the position of an object in space, and for this, you must know three dimensions, ie, x-axis, y-axis, and z-axis. Class 12 maths chapter 11 notes help you to understand the relation between two lines in space with the help of direction cosines, direction ratios. Also, these notes contain the equation of a straight line in different forms and the angle between them. In real life, 3D geometry is used in architecture, aviation, robotics, and 3D modeling.
NCERT class 12th Math chapter 11 notes contain standard formulas and detailed information that are used to solve the questions with the help of suitable examples. Our subject matter experts ensure that all concepts are covered as per the latest syllabus of CBSE in NCERT Notes for Class 12 Math chapter 11.
A line OA makes angles α,β,γ with the x,y, and z-axis, respectively. Then cosα, cosβ, and 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.
Direction cosines of ratios a,b,c :
Direction cosines pass through two points:
Direction cosines:
Direction ratios:
NOTE: Directions ratios need not be unique
Equation of Straight Line:
Equation of a Line passing through the Point and parallel to vector b
where,
Cartesian form:
Where,
l,m,n are the direction cosines, then the equation is:
Equation of line that passes through 2 points:
Points are :
Cartesian form of two points:
Points are :
Vector equations of 2 lines:
The angle between 2 lines:
Cartesian form: Let be an angle between the lines below:
Then,
Direction cosines with angles are:
Few conditions:
When lines are perpendicular,
Then Cartesian form:
When lines are parallel
Then Cartesian form:
Shortest path:
Vector equations of 2 lines:
Shortest distance
where
Cartesian form for two lines:
Cartesian form :
Shortest distance between the lines:
Distance between two Parallel Lines:
Lines are said to be coplanar when they are parallel.
Vector equations of 2 lines:
Distance between two Parallel Lines:
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 it satisfies any one of the following:
Equation of a plane in normal form:
Cartesian form: one equation of the plane is ax + by + cz = d, and another equation of the plane is lx + my + nz = p
Formula: Foot of perpendicular (ld, md, nd).
Equation of a plane perpendicular to a vector and passing through a point:
Vector equation:
Cartesian form: Equation of a plane that passes through the point
Equation of a plane passing through non-collinear points:
Vector form:
Cartesian form:
Equation:
If they are collinear:
Intercept Form:
Let a, b, and c be the x-intercept, y-intercept, and z-intercept, respectively, then
Equation of intercept: xa + by + zc = 1
Equation of Plane Passing through the intersection of two Planes: The equation of the planes is
Equation of the plane passing through the intersection :
Cartesian form: equation of planes are
Equation of the plane passing through the intersection :
Coplanarity:
Lines
Cartesian form:
For Two lines
The angle between Two Planes: θ is the angle between two planes.
Cartesian form: planes are
Angle Between Line and Plane:
Vector form: The equation of a line is
angle θ between the line and the normal to the plane is
Cartesian form: a, b, and c are direction ratios, and lx + my + nz + d = 0 is the equation of the plane.
NCERT Class 12 Maths Chapter 11 Notes |
Subject-wise NCERT Exampler solutions
NCERT problems are very important in order to perform well in exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 12 Maths Chapter 11 Three-Dimensional Geometry.
Students are advised to go through the NCERT Class 12 Maths Chapter 11 Notes before solving the questions.
To boost your exam preparation as well as for quick revision, these NCERT notes are very useful.
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