Let the unit vector orthorgonal to v = <3, 4, 0> be u = <a, b, 0>, then
3a + 4b = 0
a = -4/3b and
sqrt(a^2 + b^2) = 1
a^2 + b^2 = 1
(-4/3b) + b^2 = 1
16/9b^2 + b^2 = 1
25/9b^2 = 1
b^2 = 9/25
b = + or - 3/5
a = -4/3(3/5) or -4/3(-3/5) = -4/5 or 4/5
Therefore, required vectors are <-4/5, 3/5> and <4/5, -3/5>