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.
VectorAngle
andNorm
. $\endgroup${v1, v2, v3} /. Solve[{Norm[{v1, v2, v3}] == 1, {v1, v2, v3}.{6, 4, -2}/ Norm[{6, 4, -2}] == Cos[Pi/3], {v1, v2, v3}.{3, -4, 7} == 0, v3 > 0}, {v1, v2, v3}] // Simplify
$\endgroup$