Exercise 5

A superhero is at the origin of a two dimensional plane. The superhero has two devices, a hoverboard that moves any distance in the direction $\binom{3}{1}$ and a magic carpet that moves any distance in the direction $\binom{1}{2}$.

• An evil villain is hiding out in the plane at the point $(-5,7)$. How many hoverboard units and magic carpet units does the superhero have to move to get to the villain?

• Is there anywhere in the plane that the villain could safely hide and not be reached? If so, give one such location. If not, explain why not.

• The superhero and the villain are transported to a three dimensional space where the superhero now has three devices.

  $$\text{hoverboard: }\begin{pmatrix}-1\\ 0\\ 3\end{pmatrix}\quad\text{magic carpet: }\begin{pmatrix}2\\ 1\\ 0\end{pmatrix}\quad\text{scooter: }\begin{pmatrix}5\\ 4\\ -9\end{pmatrix}$$

  Is there anywhere that the villain could safely hide? If so, give one such location and if not, explain why not.