Finding parametrized curve

Question

Unit sphere parametrized as $x(θ, ϕ) = (\sin (θ) \cos (ϕ), \sin (θ) \sin (ϕ), \cos (θ))$, how would one find a parametric curve which has $α(0)=(1,0,0)$ and $α'(0)=(0,3,4)$?

Answer 1

Define $v_0=(1,0,0)$ and $v_1=\tfrac1{5}(0,3,4)$. Then $$ \alpha(t) = \cos(5t)v_0+\sin(5t)v_1 $$ is the curve you want.