Let us work with the standard inner product on $\mathbbmss{R}^{3}$ (dot product) so we can get a nice geometrical visualization.

Consider the three vectors

which are linearly independent (the determinant of the matrix $A=(v_{1}|v_{2}|v_{3})=116\neq 0)$ but are not orthogonal.

We will now apply Gram-Schmidt to get three vectors $w_{1},w_{2},w_{3}$ which span the same subspace (in this case, all $R^{3}$) and orthogonal to each other.

First we take $w_{1}=v_{1}=(3,0,4)$. Now,

that is,

and finally

which gives

and so $\{w_{1},w_{2},w_{3}\}$ is an orthogonal set of vectors such that $\langle w_{1},w_{2},w_{3}\rangle=\langle v_{1},v_{2},v_{3}\rangle$.

If we rather consider $\{w_{1}/\|w_{1}\|,w_{2}/\|w_{2}\|,w_{3}/\|w_{3}\|\}$ then we get an orthonormal set.