If two vectors are not parallel, can we still compare them i.e. greater than or smaller than?

Question

If vectors $x$ and vector $y$ are a member of $R^n$, and they are not parallel. Can we say anything about $x>y$ or $x<y$?

I know that they won't be equal because if $x =$ [$x_1, x_2,..., x_n$] and $y =$ [$y_1, y_2,..., y_n$]

$x_i$ will not be equal to $y_i$ for $1<=i<=n$.

Answer 1

There is no meaningful order commonly in use for vectors. The best you can do is compare magnitudes $|\mathbf x|$ and $|\mathbf y|$ since these are real numbers and comparison is meaningful.

Answer 2

The best you could do is compare their magnitudes or perhaps the sum of their entries.

Answer 3

It depends on what order you choose. Actually, we can choose a Lexicographical order to make every pair of elements is comparable:

https://en.wikipedia.org/wiki/Total_order#Orders_on_the_Cartesian_product_of_totally_ordered_sets

If you just consider the product order (https://en.wikipedia.org/wiki/Product_order), not every pair is comparable, but it is a partial order.

Generally speaking, whether $x$ and $y$ are comparable or not depends on the "rule" you define.