The document discusses the concepts of direction cosines and direction ratios in 3D geometry. It provides mathematical relations and examples to illustrate how to find these values for lines and points in space, including the conditions for collinearity of points. Additionally, it offers direction cosines for the coordinate axes and direction ratios for line segments connecting given points.

1. Physics Helpline
L K Satapathy
3D Geometry Theory 1

2. Physics Helpline
L K Satapathy
( , , ) (cos , cos , cos )l m n   
Direction Cosines of a line :
Cosines of the angles made by a line with the positive direction of X , Y and Z
axes are known as the direction cosines of the line , denoted by ( l , m , n ).
If  ,  and  be the angles made by the line with coordinate axes , then
Relation between the direction cosines : 2 2 2
1l m n  
Also
2 2 2 2 2 2
sin sin sin (1 cos ) (1 cos ) (1 cos )            
2 2 2
3 (cos cos cos )     
2 2 2
3 ( ) 3 1 2l m n      
[ Refer Theory of vectors 3 ]
3 D Geometry Theory 1

3. Physics Helpline
L K Satapathy
Direction Ratios of a line :
Direction ratios are three real numbers ( a , b , c ) , which are proportional to the
direction cosines .
. . . (1)
l m n
a b c
  
2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
1
(1)
l m n l m n
a b c a b c a b c
 
    
   
2 2 2
1l m n
a b c a b c
   
  
2 2 2 2 2 2 2 2 2
, ,
a b c
l m n
a b c a b c a b c
   
        
To find the Direction Cosines from Direction Ratios :
3 D Geometry Theory 1

4. Physics Helpline
L K Satapathy
Direction ratios of the line passing through the points
are given by
1 1 1 2 2 2( , , ) ( , , )x y z and x y z
2 1 2 1 2 1( , , ) ( , , )a b c x x y y z z   
2 1
2 2 2
2 1 2 1 2 1
( )
( ) ( ) ( )
x x
l
x x y y z z

 
     
2 1
2 2 2
2 1 2 1 2 1
( )
( ) ( ) ( )
y y
m
x x y y z z

 
     
2 1
2 2 2
2 1 2 1 2 1
( )
( ) ( ) ( )
z z
n
x x y y z z

 
     
 The Direction Cosines are
3 D Geometry Theory 1

5. Physics Helpline
L K Satapathy
Direction Cosines of the Coordinate axes :
(i) The x-axis makes angles 0 with x-axis and 90 with each of y and z-axes
(ii) The y-axis makes angles 0 with y-axis and 90 with each of x and z-axes
(iii) The z-axis makes angles 0 with z-axis and 90 with each of x and y-axes
( , , ) (cos , cos , cos )l m n   
( , , ) (cos , cos , cos )l m n   
(cos0 , cos90 , cos90 ) (1 , 0 , 0)o o o
 
(cos90 , cos0 , cos90 ) (0 , 1 , 0)o o o
 
 Direction Cosines are
 Direction Cosines are
( , , ) (cos , cos , cos )l m n   
(cos90 , cos90 , cos0 ) (0 , 0 , 1)o o o
 
 Direction Cosines are
3 D Geometry Theory 1

6. Physics Helpline
L K Satapathy
Example 1:
Show that the points A (2 , 3 , 4) , B (– 1 , – 2 , 1) and C (5 , 8 , 7) are collinear.
Ans:
Direction Ratios of AB = (– 1 – 2 , – 2 – 3 , 1 – 4 ) = (– 3 , – 5 , – 3)
Direction Ratios of BC = (5 + 1 , 8 + 2 , 7 – 1 ) = (6 , 10 , 6)
We observe that
3 5 3
6 10 6
  
 
 Direction ratios of AB and BC are proportional .  AB and BC are parallel .
And B is the common point of AB and BC .
 The points A , B and C are collinear .
3 D Geometry Theory 1

7. Physics Helpline
L K Satapathy
Example 2:
Find the direction cosines of the sides of the triangle , whose vertices are
Ans:
( 3 , 5 , – 4 ) , (– 1 , 1 , 2) and (– 5 , – 5 , – 2) .
Let A = ( 3 , 5 , – 4 ) , B = (– 1 , 1 , 2) and C = (– 5 , – 5 , – 2) .
Direction Ratios of AB = (– 1 – 3 , 1 – 5 , 2 + 4 ) = (– 4 , – 4 , 6)
Direction Cosines of AB
4 4 6
, ,
16 16 36 16 16 36 16 16 36
  
  
      
4 4 6 2 2 3
, , , ,
68 68 68 17 17 17
      
    
   
3 D Geometry Theory 1

8. Physics Helpline
L K Satapathy
Direction Ratios of BC = (– 5 + 1 , – 5 – 1 , – 2 – 2 ) = (– 4 , – 6 , – 4 )
Direction Cosines of BC
4 6 4
, ,
16 36 16 16 36 16 16 36 16
   
  
      
4 6 4 2 3 2
, , , ,
68 68 68 17 17 17
        
    
   
Direction Ratios of CA = (3 + 5 , 5 + 5 , – 4 + 2 ) = (8 , 10 , – 2 )
Direction Cosines of CA
8 10 2
, ,
64 100 4 64 100 4 64 100 4
 
  
      
8 10 2 4 5 1
, , , ,
168 168 168 42 42 42
   
    
  
3 D Geometry Theory 1

9. Physics Helpline
L K Satapathy
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