How can i easily find two 3-dimensional vectors that are 45 degrees apart?

Question

I'm supposed to give two 3-diemensional vectors that are parallel to the x,y plane and have a 45 degrees angle between them and go through the point P(1|1|0). By using the arccos(x) method,my x is supposed to be 1/sqrt(2) but i can't find two vectors that will give me 1/sqrt(2).

Answer 1

Point P(1,1,0) in x,y plane. Let take one vector parallel x-axis, if initial point is P(1,1,0) then terminal point is Q(2,1,0) and we have vector PQ = <2 - 1, 1 - 1, 0 - 0> = <1, 0, 0> = i,

Second vector is parallel line y = x, so if I take point R(2,2,0) then vector PR = <2 - 1, 2 - 1, 0 - 0> = <1, 1, 0> =

i + j