Question 7
In $\triangle ABC,$ $\angle ABC=90^\circ$ and $BA=BC=\sqrt2.$ Points $P_1,$ $P_2,$ $\ldots,$ $P_{2024}$ lie on hypotenuse $AC$ so that $AP_1=P_1P_2$ $=P_2P_3$ $=\cdots$ $=P_{2023}P_{2024}$ $=P_{2024}C.$
What is the length of the vector sum $$
\begin{aligned} &\vec{BP_1}+\vec{BP_2}+\vec{BP_3} \\ &\quad {}+\cdots+\vec{BP_{2024}}? \end{aligned}
$$
What is the length of the vector sum $$
\begin{aligned} &\vec{BP_1}+\vec{BP_2}+\vec{BP_3} \\ &\quad {}+\cdots+\vec{BP_{2024}}? \end{aligned}
$$