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Direction Cosines and Direction Ratios of a Line

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Introductiondir_ratio_2

We have already learned the basic concepts of vectors.  In this topic we will use the concepts of vector algebra to the three dimensional geometry.  In the three dimensional geometry, we deal with direction cosines, direction ratios, equations of line in space, equation of plane in space etc.



dir_ratio_1

Direction Cosines

If a directed line L’ passing through the origin makes angles α, β and γ with x, y and z axes respectively then cosine of these angles namely,

cosα, cosβ and cosγ are called direction cosines of the directed line L’.

Usually the direction cosines are denoted by l, m and n


l=cosα, m=cosβ and n=cosγ



Relation between the direction cosines of a line

If l, m and n are the direction cosines of a line then l2+m2+n2=1

Also, cos2α+cos2β+cos2γ=1



Direction cosines of a line passing through two points

Let P(x1, y1, z1) and Q(x2, y2, z2) be two points on a line L, then

PQ = √((x2-x1)2+(y2-y1)2+(z2-z1)2
)

Direction cosines of the line L is given by,
 dir_ratio_6
                            


Direction Ratios of a line

Any three numbers which are proportional to the direction cosines of a line are called direction ratios of the line.  If l, m and n are direction cosines abd a, b and c are direction ratios of a line then a=λl, b=λm and c=λn.

It can also be written as dir_ratio_4


If P(x1, y1, z1) and Q(x2, y2, z2) are any two points the direction ratios of PQ is given by <x2-x1, y2-y1, z2-z1>



Direction cosines of x, y and z-axis

X-axis makes angles 0˚, 90˚ and 90˚ with itself, so the direction cosines are cos0˚, cos90˚ and cos90˚ = <1, 0, 0>

Y-axis makes angles 90˚, 0˚ and 90˚ with itself, so the direction cosines are cos90˚, cos0˚ and cos90˚ = <0, 1, 0>


Z-axis makes angles 90˚, 90˚ and 0˚ with itself, so the direction cosines are cos90˚, cos90˚ and cos0˚ = <0, 0, 1>



Condition for collinearity

If a1, b1, c1 and a2, b2, c2 are the direction cosines of line joining two points then the points are said to be collinear

if dir_ratio_5

                                                          

Example:
Find the direction cosines of a line which makes equal angles with the coordinate axes.


Solution:
Given α = β = γ, so cosα = cosβ = cosγ


l=m=n


l2+m2+n2=1


l2+l2+l2=1


3l2=1


l2=1/3


l=±1/√3


l=m=n=±1/√3


Hence direction cosines are <±1/√3, ±1/√3, ±1/√3>



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