# NCERT Class 12-Mathematics: Chapter –11 Three Dimensional Geometry Part 1

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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