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CLASS –XII, LESSON NOTES
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. Represented by l, m, n

 If l,m,n are the direction cosines of a line, then

 Direct ion cosines of a line joining two points and
are

where

 Direction ratios of a line are the numbers which are proportional to the
direct ion cosines of a line,. represented by a,b,c

 If l, m,n are the direct ion cosines and a,b,c 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.

 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 (x 1,y 1,z 1) and having direct ion
cosines l, m ,n is