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**: - l^{2}+m^{2}+n^{2}=1 i.e. cos^{2}α + cos^{2} β+ cos^{2} 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=√(a^{2}+b^{2}+c^{2})

IPQ^{-->}I^{2}=(a^{2}+b^{2}+c^{2})

=r^{2} (cos^{2} α +cos^{2}β +cos^{2}γ)= (a^{2}+b^{2}+c^{2}) =r^{2} (cancelling r^{2} from both the sides).

=**(****cos ^{2} α + cos^{2}**

**l ^{2}+m^{2}+n^{2} =1**.

The above proof holds good for any line.

Hence Proved.

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