Find the equations of the two lines through the origin which intersect the line `(x-3)/2=(y-3)/1=z/1` at angle of `pi/3` each.

Explanation

Given equation of the lines `(x-3)/2=(y-3)/1=z/1=lamda`

So, DR's of the line are 2,1,1 and DC's of the given line are `2/(sqrt(6)),1/(sqrt(6)),1/(sqrt(16))`
Also, the required lines make angle `(pi)/3` with the given line.
From eq. i `x=(2lamda+3),y=(lamda+3)` and `z=lamda`
`:' cos theta=(a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2))/(sqrt(a_(1)^(2)+b_(1)^(2)+c_(1)^(2)sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))`
`"cos" (pi)/3=((4lamda+6)+(lamda+3)+(lamda))/(sqrt(6)sqrt((2lamda+3)^(2)+(lamda+3)^(2)+lamda^(2))`
`implies1/2=(6lamda+9)/(sqrt(6)sqrt((4lamda^(2)+9+12lamda+lamda^(2)+9+6lamda+lamda^(2)))`
`implies(sqrt(6))/2=(6lamda+9)/(sqrt(6lamda^(2)+18lamda+18))`
`implies 6sqrt((lamda^(2)+3lamda+3))=2(6lamda+9)`
`implies 36(lamda^(2)+3lamda+3)=36(4lamda^(2)+9+12lamda)`
`implies lamda^(2)+3lamda+3=4lamda^(2)+9+12lamda`
`implies 3lamda^(2)+9lamda+6=0`
`implies lamda^(2)+3lamda+2=0`
`= lamda(lamda+2)+1(lamda+2)=0`
`implies lamda(lamda+2)+1(lamda+2)=0`
`implies (lamda1)(lamda+2)=0`
`:. lamda=-1,-2`
So, theEC are `1,2,-1` and `-1,1,-2`.
Also both the required lines passes through origin
So, the equations of required lines are `x/1=y/2=z/(-1)` and `x/(-1)=y/1=z/(-2)`