Given a unit vector $u\in \mathbb R^n$, take $v$ to be sampled uniformly at random from the unit sphere centered at $0$ in $\mathbb R^n$. Is the probability of $u$ and $v$ being orthogonal equal to zero?
I am inclined to say yes, because the measure of a hyperplane is zero. Is this sufficient (and correct)?
You are right the probabily will be $0$. The uniform probability distrubution means that we the probability distribution is defined by the Lebesgue measure inherited by $\mathbb S^{n-1}$ from $\mathbb R^n$. For any measurable $A \subset \mathbb S^{n-1}$ is given by $$P(A) = \frac{\mu(A)}{\mu(\mathbb S^{n-1})} $$ If $\mu(A)=0$ (like in the case of the great circle formed by vectors $v \perp u$), then also $P(A)=0$.