Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem
\[d_{1}=\min _{a_{1} \in \mathbb{R}}\left\|\vec{u}-a_{1} \vec{v}_{1}\right\|,\]
where we denote the length $\sqrt{\vec{w}^{T} \vec{w}}$ of a vector $\vec{w} \in \mathbb{R}^{n}$ by $\|\vec{w}\|$. Let the pair of real numbers $\left(b_{1}^{*}, b_{2}^{*}\right)$ be such that they solve the following optimization problem
\[d_{2}=\min _{b_{1}, b_{2} \in \mathbb{R}}\left\|\vec{u}-\left(b_{1} \vec{v}_{1}+b_{2} \vec{v}_{2}\right)\right\|.\]
Let
\[\begin{array}{l}
\vec{p}_{1}=a_{1}^{*} \vec{v}_{1} \\
\vec{p}_{2}=b_{1}^{*} \vec{v}_{1}+b_{2}^{*} \vec{v}_{2}
\end{array}\]
Compute $\left(\vec{p}_{2}-\vec{p}_{1}\right)^{T} \vec{p}_{1}$.
- $\left\|\vec{u}-\vec{p}_{1}\right\|$
- $\left\|\vec{u}-\vec{p}_{2}\right\|$
- $\left\|\vec{u}-\vec{p}_{2}-\vec{p}_{1}\right\|$
- $\left\|\vec{u}-\left(\vec{p}_{2}-\vec{p}_{1}\right)\right\|$
- $0$