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)

where r=√(a2+b2+c2)

IPQ-->I2=(a2+b2+c2)

=r2 (cos2 α +cos2β +cos2γ)= (a2+b2+c2) =r2 (cancelling r2 from both the sides).

=(cos2 α + cos2β + cos2γ) =1

l2+m2+n2 =1.

The above proof holds good for any line.

Hence Proved.

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