IB-07Direction Ratios and Direction Cosines(29-35)

8/9/2019 IB-07Direction Ratios and Direction Cosines(29-35)

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7. DIRECTI ON RAT IOS AN D DIRECTION COSINES

Synopsis :

1. If a directed line (ray) makes angle a, b, g with positive directions of the axes of x,y and zrespectively the cosa, cosa, cosg are called the direction cosines (d.cs) of the line and these areusually denoted by l,m,n.

2. A line in a space can be extended in two directions, it has two sets of direction cosines. If ( ), ,l m n is one set then other set is . Here ( ), ,l m n

i) 2 2 2 1l m n+ + =

ii) 2 2 2cos cos cos 1 + + =

iii) 2 2 2sin sin sin 2 + + =

3. Direction cosines of i) x axis are (1,0,0)

ii) y axis are (0,1,0)

iii) z axis are (0,0,1)

4. If (l, m, n) are direction cosines of a line OP where 'O' is the origin and OP = r then thecoordinates P are (lr, mr, nr).

5. If the coordinates of a point P are (x,y, z) and OP=r then the direction Cosines of OP are(x/r, y/r/ z/r).

6. If ( )1 1 1, , A x y z= and ( )2 2 2, , B x y z= then AB d.cs1 2 1 2 1 2

, , x x y y z z

AB AB AB

are

7. If is the angle between the two lines whose d.cs are 1 1 1, ,l m n and 2 2 2, ,l m n then

i) 1 2 1 2 1 2cos l l m m n n = + +

ii) [ ]21 2 2 1sin l m l m =

iii) if the lines are perpendicular 1 2 1 2 1 2 0l l m m n n+ + = then

iv) If the lines are parallel then 1 1 12 2 2

l m nl m n

= = .

8. If are the d.cs of the line AB and a,b,c are three numbers such thatl m na b c

= = then are called

the direction ratios (d.rs) of the line AB.

9. If a,b,c are the d.rs of the line AB then the d.cs of the line AB are

2 2 2 2 2 2 2 2 2, ,

a b c

a b c a b c a b c+ + + + + +

(or)

2 2 2 2 2 2 2 2 2, ,

a b c

a b c a b c a b c

+ + + + + +

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10. If ( )1 1 1, , A x y z= and ( )2 2 2, , B x y z= then the d.rs of AB are 1 2 1 2 1 2, , x x y y z z

11. If is the angle between the two lines whose d.rs are 1 1 1, ,a b c and 2 2 2, ,a b c then

i) 1 2 1 2 1 22 2 2 2 2 21 1 1 2 2 2

cos a a b b c ca b c a b c

+ +

= + + + +

ii)( )21 2 2 1

2 2 2 2 2 21 1 1 2 2 2

sina b a b

a b c a b c

=

+ + + +

iii) if the lines are perpendicular 1 2 1 2 1 2 0a a b b c c+ + = then

iv) if the lines are parallel then 1 1 12 2 2

a b ca b c

= = (or) ( ) ( )1 1 1 2 2 2, , , ,a b c k a b c= where k is a

constant. Where { }0k R

12. If 1 1 1, ,a b c and 2 2 2, ,a b c are the d.rs of the lines AB and AC then the d.rs of the lineperpendicular to both AB and AC are 1 2 2 1 1 2 2 1 1 2 2 1, ,b c b c c a c a a b a b .

13. If 1 1 1, ,l m n and 2 2 2, ,l m n are the d.cs of the lines AB and AC then the d.rs of the bisectorsof the angles between AB and AC are 1 2 1 2 1 2, ,l l m m n n+ + + and 1 2 1 2 1 2, ,l l m m n n

14. If is the angle between two lines whose d.cs are 1 1 1, ,l m n and 2 2 2, ,l m n then the d.cs of their

angular bisectors are 1 2 1 2 1 2, ,2cos 2cos 2 cos

2 2 2

l l m m n n

+ + +

And 1 2 1 2 1 2, ,2sin 2sin 2sin

2 2 2

l l m m n n

15. The projection of a point on a line is the foot of the perpendicular from that point to the line.

16. If ( )1 1 1, , A x y z= and ( )2 2 2, , B x y z= then the projection of AB on the line whose d.cs are , , l m n is ( ) ( ) ( )1 2 1 2 1 2l x x m y y n z z + +

17. If ( )1 1 1, , A x y z= and ( )2 2 2, , B x y z= then

i) the projection of AB on the x-axis is 1 2 x x

ii) the projection of AB on the y-axis is 1 2 y y

iii) the projection of AB on the z-axis is 1 2 z z

18. If ( )1 1 1, , A x y z= and ( )2 2 2, , B x y z= then

i) the projection of AB on the yoz plane is ( ) ( )2 21 2 1 2 y y z z +

ii) the projection of AB on the zox plane is ( ) ( )2 21 2 1 2 z z x x +




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iii) the projection of AB on the xoy plane is ( ) ( )2 21 2 1 2 x x y y +

19. i) If the projections of a line of length d on the coordinate axes are 1 2 3, &d d d

respectively, then2 2 2 2

1 2 3d d d d + + = ii) If the projections of a line of length d on the coordinate planes are 1 2 3, &d d d

respectively, then 2 2 2 21 2 3 2d d d d + + =

20. The d.rs of two lines are given by the equations 0al bm cn+ + = and 0 fmn gnl hlm+ + = . If the

two lines are perpendicular then 0 f g ha b c

+ + = . If the two lines are parallel then

0af bg ch+ + =

21. The d.rs of two lines are given by the equations 0al bm cn+ + = and 2 2 2 0ul vm wn+ + = . If the

two lines are perpendicular then ( )( )2 0a v w + = . If the two lines are parallel then2 2 2

0a b cu v w

+ + =

22. The equation of the line joining the points ( )1 1 1, , x y z and ( )2 2 2, , x y z is

1 1 1

2 1 2 1 2 1

x x y y z zt

x x y y z z

= = = where t is a parameter. Here any point on the line is

( ) ( ) ( )1 2 1 1 2 1 1 2 1, , x t x x y t y y z t z z+ + +

23. If , ,a b c are the d.rs of the line passing through the point( )1 1 1

, , x y z then the equation of that

line is 1 1 1 x x y y z z

a b c

= =

24. Lagranges identity: If 1, 1 1 2 2 2, , , ,l m n l m n are real numbers then ( )2 2 21 1 1l m n+ + ( )2 2 22 2 2l m n+ +

( )21 2 1 2 1 2l l m m n n + +

( ) ( ) ( )2 2 21 2 2 1 1 2 2 1 1 2 2 1m n m n n l n l l m l m= + +

25. Angle between any two diagonals of a cube is 11

cos3

26. Angle between diagonal of a cube and diagonal of a face of a cube is

27. If , , , are angles made by a ray with four diagonals of a cube, then

2 2 2 2 4cos cos cos cos3

+ + + =

28. If a line makes equal angles with the coordinate axes, then the line make an angle 11

cos3

or ( )1tan 2 or 1 2sin 3



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IB-07Direction Ratios and Direction Cosines(29-35)