# NCERT Class 12-Mathematics: Chapter – 11 Three Dimensional Geometry Part 1 (For CBSE, ICSE, IAS, NET, NRA 2023)

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## 11.1 Overview

**11.1. 1** Direction cosines of a line are the cosines of the angles made by the line with positive directions of the co-ordinate axes.

**11.1. 2** If *l*, *m*, *n* are the direction cosines of a line, then

**11.1. 3** Direction cosines of a line joining two points are

Where

**11.1. 4** Direction ratios of a line are the numbers which are proportional to the direction cosines of the line.

**11.1. 5** If are the direction cosines and are the direction ratios of a line, then

**11.1. 6** Skew lines are lines in the space which are neither parallel nor intersecting. They lie in the different planes.

**11.1. 7** 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.

**11.1. 8** If are the direction cosines of two lines and is the acute angle between the two lines, then

**11.1. 9** If and are the directions ratios of two lines and is the acute angle between the two lines, then

**11.1. 10** Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector is .

**11.1. 11** Equation of a line through a point and having directions cosines (or, direction ratios and ) is

**11.1. 12** The vector equation of a line that passes through two points whose positions vectors are and is

**11.1. 13** Cartesian equation of a line that passes through two points and is

**11.1. 14** If is the acute angle between the lines and , then is given by or .

**11.1. 15** If are equations of two lines, then the acute angle between the two lines is given by

**11.1. 16** The shortest distance between two skew lines is the length of the line segment perpendicular to both the lines.

**11.1. 17** The shortest distance between the lines is

**11.1. 18** Shortest distance between the lines: and

**11.1. 19** Distance between parallel lines is

**11.1. 20** The vector equation of a plane which is at a distance from the origin, where is the unit vector normal to the plane, is

**11.1. 21** Equation of a plane which is at a distance *p* from the origin with direction cosines of the normal to the plane as .

**11.1. 22** The equation of a plane through a point whose position vector is and perpendicular to the vector is , where

**11.1. 23** Equation of a plane perpendicular to a given line with direction ratios and passing through a given point is

**11.1. 24** Equation of a plane passing through three non-collinear points , and is

**11.1. 25** Vector equation of a plane that contains three non-collinear points having position vectors is

**11.1. 26** Equation of a plane that cuts the co-ordinates axes at , and is

**11.1. 27** Vector equation of any plane that passes through the intersection of planes and is where is any non-zero constant.

**11.1. 28**Cartesian equation of any plane that passes through the intersection of two given planes is

**11.1. 29** Two lines are coplanar if

**11.1. 30** Two lines are coplanar if

**11.1. 31** In vector form, if is the acute angle between the two planes, and

**11.1. 32**The acute angle between the line and plane is given by