|Class 12 Maths Three Dimensional Geometry||Direction Cosines of a Line|
Direction Cosines of a Line
Consider a line passing through origin and making angle α with x-axis, β angle with y-axis and γ angle with z-axis,then cosα,cosβ and cosγareknown as direction cosines of the line.
Direction cosines are the cosine of the angles which a line makes with XY and Z-axis.
They are denoted by(l,mand n).Therefore, cos α =l,cosβ=m and cos γ= n.
To prove: - l2+m2+n2=1 i.e. cos2α + cos2 β+ cos2 γ =1.
Proof:- Consider a line PQ and its equation is given as:-
PQ->= (a î+bĵ+ck̂)
=r(cos α î+ cosβ ĵ+ cos γ k̂)
=r2 (cos2 α +cos2β +cos2γ)= (a2+b2+c2) =r2 (cancelling r2 from both the sides).
=(cos2 α + cos2β + cos2γ) =1
The above proof holds good for any line.