Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors such that $\vec{v}_{1}^{T} \vec{v}_{2}=0$. 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
\[\begin{aligned}
\vec{v}_{*} &=a_{1}^{*} \vec{v}_{1}+a_{2}^{*} \vec{v}_{2} \\
\vec{v} &=a_{2}^{*} \vec{v}_{1}+a_{1}^{*} \vec{v}_{2} .
\end{aligned}\]
Compute the inner product $\vec{v}_{*}^{T} \vec{v}$.
- $\vec{u}^{T} \vec{v}$
- $\|\vec{u}\|^{2}-\left\|\vec{v}_{*}\right\|^{2}$
- $\left\|\vec{v}_{*}\right\|^{2}-\|\vec{u}\|^{2}$
- $0$
- None of the above