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.

11.1.28Cartesian 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

11.1.32The acute angle between the line and plane is given by