I'm given the following problem:
Consider the curve $\theta = t, \phi = 0$ on the unit sphere. At the point $t = \pi/4$, the tangent vector to the curve is $(1,0)$. A tangent vector of the form $(0, a)$ will be perpendicular to the previous tangent vector at this point. Given these 2 conditions, I have been asked to find a line/geodesic that passes this point such that such that the tangent vector of the geodesic is perpedicular to the tangent vector of the original curve at this point.
Is there a way to go about this problem without having to explicitly solve the geodesic equation?
