Question

State the cartesian equations of two lines that make angle of 60 degrees with the line r=(1,3)+s(1,1)

Answer 1

direction of the given line is (1,1)
so if Ø is the angle it makes with the x-axis, then
tanØ = 1/1
Ø = 45°

so a line which makes an angle of 60° with the given line would have an angle of (45°+60°) or 105° or -15°

now tan 105°
= -tan75°
= -tan(45+30)
= -(tan45 + tan30)/(1 - tan45tan30)
= -(1+ 1/√3)/(1-(1)(1/√3)
=-(√3 + 1)/(√3-1)

so a possible linear equation could be
(√3+1)x + (√3-1)y = 55 , (or any other constant)

I will leave the finding of the other line up to you,
you will need
tan (-15°)

hint: tan -15 = tan(30-45)