20. Orthonormal Basis from Scratch

Let $v_1,v_2,v_3$ be three linearly independent vectors in $\mathbb{R}^3$.

Derive a general procedure for constructing an orthogonal basis $u_1,u_2,u_3$ for $\operatorname{span}\{v_1,v_2,v_3\}$ using the standard inner product.

Your procedure should be expressed in terms of $v_1,v_2,v_3$, and you must prove that the resulting vectors are mutually orthogonal and span the same subspace.