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Question
If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.
Solution
Let `alpha,beta,gamma ` be the angles made by the line with X-, Y-, Z- axes respectively.
`l=cosalpha, m=cosbeta and n=cosgamma`
Let `bara=a_1hati+a_2hatj+a_3hatk` be any non-zero vector along the line.
Since `hati` is the unit vector along X-axis,
`bara.hati=|bara|.|hati|cosalpha=acosalpha`
Also, `bara.hati=(a_1hati+a_2hatj+a_3hatk).hati`
`=a_1xx1+a_2xx0+a_3xx0=a_1`
`acosalpha=a_1` ..............................(1)
Since `hatj` is the unit vector along Y-axis,
`bara.hatj=|bara|.|hatj|cosbeta=acosbeta`
`bara.hatj=(a_1hati+a_2hatj+a_3hatk).hatj`
`=a_1xx0+a_2xx1+a_3xx0=a_2`
`acosbeta=a_2` ......................(2)
similarly `acosgamma=a_3` .............(3)
from equations (1), (2) and (3),
`a^2cos^2alpha+a^2cos^2beta+a^2cos^2gamma=a_1^2+a_2^2+a_3^2`
`a^2(cos^2alpha+cos^2beta+cos^2gamma)=a^2 ` `[a=|bara|=sqrt(a_1^2+a_2^2+a_3^2)]`
`therefore cos^2alpha+cos^2beta+cos^2gamma=1`
`i.e l^2+m^2+n^2=1`
also
`alpha=?, beta=135^@,gamma=45^@`
`cos^2alpha+cos^2beta+cos^2gamma=45^@`
`cos^2alpha+cos^2 135^@+cos^2 45^@=1`
`cos^2alpha+1/2+1/2=1`
`cos^alpha=0`
`therefore alpha=pi/2 or (3pi)/2`
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