mathematica finding unknown vector that satisfies conditions

Question

How do you write each condition as an equation or inequality to find a vector $v$ that satisfies

$v$ is a unit vector: $||v||=1$, $v$ forms an angle of π/3 with the vector $(6,4,-2)$, $v$ is orthogonal to the vector $(3,-4,7)$, the third component of $v$ is positive.

Answer 1

Something like this:

vec = {x, y, z};
v1 = {3, -4, 7};
v2 = {6, 4, -2};

eqs = {Norm[vec] == 1, vec.v1 == 0, vec.(v2/Norm[v2]) == Cos[Pi/3], z > 0};

Solve[eqs, {x, y, z}] // N

(* {{x -> 0.176786, y -> 0.885294, z -> 0.430117}} *)