## Class 12 Maths Revision Notes for Three Dimensional Geometry of Chapter 11 – Free PDF Download

Free PDF download of Class 12 Maths revision notes & short key-notes for Three Dimensional Geometry of Chapter 11 to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books.

**CBSE Class 12 Mathematics Revision Notes Chapter 11 Three Dimensional Geometry**

- Direction cosines of a line: Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.
- are the direct ion cosines of a line, then
- Direct ion cosines of a line joining two pointsandare

· where

- Direction ratios of a line are the numbers which are proportional to the direct ion cosines of a line.
- If are the direct ion cosines and are the direct ion ratios of a line

Then, ,,

**Skew lines**: Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.**Angle between two skew lines**: Angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines.- If are the direction cosines of two lines; and is the acute angle between the two lines; then,

- Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector
- Equation of a line through a point and having direct ion cosines is
- The vector equation of a line which passes through two points whose position vectors are
- Cartesian equation of a line that passes through two points and is
- If is the acute angle between and then,
- If and are the equations of two lines, then the acute angle between the two lines is given by
- Shortest distance between two skew lines is the line segment perpendicular to both the lines.
- Shortest distance between and is
- Shortest distance between the lines: and is

- Distance between parallel lines and is
- In the vector form, equation of a plane which is at a distance d from the origin, and is the unit vector normal to the plane through the origin is
- Equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane as l, m, n is .
- The equation of a plane through a point whose position vector is a and perpendicular to the vector is .
- Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point is
- Equation of a plane passing through three non collinear points ,
- Vector equation of a plane that contains three non collinear points having position vectors and is .
- Equation of a plane that cuts the coordinates axes at is .
- Vector equation of a plane that passes through the intersection of planes and is , where is any non-zero constant.
- Cartesian equation of a plane that passes that passes through the intersection of two given planes and is
- Two lines and are coplanar if
- Two planes and are coplanar if
- In the vector form, if is the angle between the two planes, and , then
- The angle between the line and the plane is
- The angle between the planes and is given by
- The distance of a point whose position vector is from the plane is .
- The distance from a point to the plane Ax + By + Cz + D = 0 is
- Equation of any plane that is parallel to a plane that is parallel to a plane Ax + By + Cz + D = 0 is Ax + By + Cz + k = 0, where k is a different constant other than D.

**CBSE Class 12 Revision Notes and Key Points**

Three Dimensional Geometry class 12 Notes Mathematics. CBSE quick revision note for class-12 Chemistry Physics Maths, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.