Consider the vectors:

${a}_{1}=\left(\begin{array}{c}0\\ -1\\ 1\\ 0\end{array}\right),{a}_{2}=\left(\begin{array}{c}0\\ 0\\ -1\\ 1\end{array}\right),{a}_{3}=\left(\begin{array}{c}2\\ 0\\ 0\\ 1\end{array}\right)$

Find a single vector $p$ which maximizes $p{a}_{i}$ for $i=1,2,3$.

To put this in context this is an economics profit max problem where p is a price and each component of the above vectors represents the quantity of the good.

I honestly have no idea how to find this p vector. It doesn't even seem possible to me that a single vector can maximize these three vectors.

${a}_{1}=\left(\begin{array}{c}0\\ -1\\ 1\\ 0\end{array}\right),{a}_{2}=\left(\begin{array}{c}0\\ 0\\ -1\\ 1\end{array}\right),{a}_{3}=\left(\begin{array}{c}2\\ 0\\ 0\\ 1\end{array}\right)$

Find a single vector $p$ which maximizes $p{a}_{i}$ for $i=1,2,3$.

To put this in context this is an economics profit max problem where p is a price and each component of the above vectors represents the quantity of the good.

I honestly have no idea how to find this p vector. It doesn't even seem possible to me that a single vector can maximize these three vectors.