  In this article, we discuss the relationship between Birkhoff-James
orthogonality of elementary tensors in the space
$L^{p}(\mu)\otimes^{\Delta_{p}}X,\; (1\leq p<\infty)$ with the individual
elements in their respective spaces, where $X$ is a Banach space whose norm is
Fr$\acute{e}chet$ differentiable and $\Delta_{p}$ is the natural norm induced
by $L^{p}(\mu,X)$. In order to study the said relationship, we first provide
some characterizations of Birkhoff-James orthogonality of elements in the
Lebesgue-Bochner space $L^{p}(\mu,X)$.
