Use of orthogonal vectors in lattice-based cryptography In lattice-based cryptography, given the basis of the lattice we compute the orthogonal vectors using Gram-Schmidt Orthogonalization process. What is the use of orthogonal vectors in lattices?
Let $\left\{\vec{b}_1,\ldots,\vec{b}_n\right\}$ ba a lattice basis and $\left\{\hat{\vec{b}}_1,\ldots,\hat{\vec{b}}_n\right\}$ be its Gram-Schmidt orthogonolization. Here are some reasons:


It helps computing a reduced basis (an equivalent basis with shorter vectors) using LLL or BKZ algorithm,
The span of $\{\vec{b}_1,\ldots,\vec{b}_i\}$ is equal to the span of $\{\hat{\vec{b}}_1,\ldots,\hat{\vec{b}}_i\}$ for $1\leq i\leq n$,
It helps you finding the volume of the lattice by multiplying the norms of $\hat{\vec{b}}_i$'s rather than computing a determinant.
It provides you upper and lower bounds on successive minimas and smoothing parameter of the lattice.