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.

Answer 1

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`