Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such that they solve the following optimization problem
\[d=\min _{a_{1}, a_{2} \in \mathbb{R}}\left\|\vec{u}-\left(a_{1} \vec{v}_{1}+a_{2} \vec{v}_{2}\right)\right\|,\]
where for a vector $\vec{w} \in \mathbb{R}^{n}$ we denote its length by $\|\vec{w}\|$. Let $\vec{v}_{*}=a_{1}^{*} \vec{v}_{1}+a_{2}^{*} \vec{v}_{2}$, so that $d=\left\|\vec{u}-\vec{v}_{*}\right\|$. Which of the following is equal to $d^{2}$ ?
- $\left\|\left.\vec{v}_{1}\right|^{2}+\right\| \vec{v}_{2}\left\|^{2}-\right\| \vec{u} \|^{2}$
- $\left\|\left.\vec{u}\right|^{2}-\right\| \vec{v}_{1}\left\|^{2}-\right\| \vec{v}_{2} \|^{2}$
- $\|\vec{u}\|^{2}-\left\|\vec{v}_{*}\right\|^{2}$
- $\left\|\vec{v}_{*}\right\|^{2}-\|\vec{u}\|^{2}$
- None of the above