Find the direction angles of the line with the X-axis which makes direction angles of `135^(@)` and `45^(@)` with Y-axes Z-axes respectively.

Explanation

Let l,m,n be the direction cosines ofline L.
Let `p(x,y,z)` be any point on line L such that l `(OP)=r`
`:. Barr=bar(OP)=xhatiyhatj+zhatk`
If `alpha,beta, gamma` be the direction angles of the line OP, then `l = cos alpha, m= cos beta, n= cos gamma`
Now, `bar(OP)*hati=(xi+yj+2zk)*i=x " "...(1)`
Also, `bar(OP)* hati=|bar(OP)|.cos alpha= r cos alpha" ".....(2)`
`:.` From (1) and (2), `x= r cos alpha`
Similarly we have `y=r cos beta and z=r cos gamma.`
Consider `x^(2)+y^(2)+z^(2)=r^(2)(cos^(2) alpha+cos^(2) beta + cos^(2) gamma)`
`:. r^(2)=r^(2)(l^(2)+m^(2)+n^(2))`
`:. l^(2)+m^(2)+n^(2)=1`
Given : `beta =135^(@), gamma= 45^(@), alpha?`
We have `cos^(2)+cos^(2)(135^(@))+cos^(2)(45^(@))=1`
`:. os^(2) alpha+ (-(1)/(sqrt(2)))^(2)+((1)/(sqrt(2)))=1`
`:. cos^(2) alpha(1)/(2)+(1)/(2)=1`
`:. cos^(2)alpha=0`
`rArr alpha=90`
Thus, the direction angle of the line with X-axis is `90^(@)`